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http://www.archive.org/details/elementarytreatiOOricerich 


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AN 


ELEMENTARY    TREATISE 


ON    THE 


DIFFERENTIAL  CALCULUS 


FOUNDKD     ON    THE 


METHOD  OF  RATES  OR  FLUXIONS 


BY 

JOHN    MINOT    RICE 

PROFESSOR    OF    MATHEMATICS    IN    THE     UNITED    STATES    NAVY 
AND 

WILLIAM    WOOLSEY    JOHNSON 

FKOFESSOR    OF    MATHEMATICS    IN     SAINT    JOHN's    COLLEGE    ANNAPOLIS    MARYLAND 


ABRIDGED   EDITION 

THIRD   THOUSAND. 

NEW   YORK: 
JOHN    WILEY    AND    SONS, 

.53  East  Tenth  Street, 
1S93. 


^^L.J~iA^   X^jt^ 


Copyright,  1880, 
John  Wiley  and  Sons. 


^^C/t^f^--^^  /';^ 


Now  York  :  J.  J.  Uttle  &  Co.,  Prlntew, 
10  to  30  Astor  Flace. 


PREFACE 


In  preparing  this  abridgment  of  their  treatise  on  the  Dif- 
ferential Calculus,  the  authors  have  endeavored  to  adapt  it 
to  the  wants  of  those  instructors  who  find  the  larger  work 
too  extensive  for  the  time  allotted  to  this  subject. 

J.  M.  R. 

W.  W.  J. 

Annapolis,  Maryland, 

Augtisif  1880. 


ivi57?C 


^  / 


CONTENTS. 


CHAPTER  I. 

r 

Functions,  Rates,  and  Derivatives. 
I. 

PAGE 

Functions   I 

Implicit  functions 3 

Inverse  functions 4 

Classification  of  functions 4 

Expressions  involving  an  unknown  function 5 

Examples  1 6 

II. 

Rates 9 

Constant  rates 10 

Variable  velocities ii 

Illustration  by  means  of  Attwood's  machine 11 

The  measure  of  a  variable  rate 12 

Differentials 12 

The  differentials  of  polynomials 13 

The  differential  of  vix 14 

Examples  II 15 

III. 

The  differentials  of  functions 16 

The  derivative — its  value  independent  of  dx 17 

The  geometrical  meaning  of  the  derivative 19 

Examples  III 21 

CHAPTER   II. 

The  Differentiation  of  Algebraic  Functions. 

IV. 

The  square  23 

The  square  root 25 

Examples  IV 26 


CONTENTS. 


V. 


PAGE 


The  product 29 

The  reciprocal 30 

The  quotient 31 

The  power ~32~ 

Examples  V 34 

CHAPTER    III. 
The  Differentiation  of  Transcendental  Functions. 

VI. 

The  logarithm 37 

The  Napierian  base 39 

The  logarithmic  curve  (jv  —  log^jr) 40 

Logarithmic  differentiation 41 

Differentials  of  algebraic  functions  deduced  by  logarithmic  differentiation 42 

Exponential  functions 43 

Examples  VI 44 

VII. 

Trigonometric  or  circular  functions ; 47 

The  sine  and  the  cosine 48 

The  tangent  ar^d  the  cotangent 49 

The  secant  and  the  cosecant 50 

The  versed  sine 50 

Examples  VII 51 

VIII. 

Inverse  circular  functions— their  primary  values 54 

The  inverse  sine  and  the  inverse  cosine 56 

The  inverse  tangent  and  the  inverse  cotangent 57 

The  inverse  secant  and  the  inverse  cosecant 58 

The  inverse  versed-sine 59 

Examples  involving  trigonometric  reductions 59 

Examples  VIII 60 

IX. 

Differentials  of  functions  of  two  variables 62 

Examples  IX 64 

Miscellaneous  examples  of  differentiation 65 


vi  CONTENTS. 


CHAPTER    IV. 
Successive  Differentiation. 


Velocity  and  acceleration 67 

Component  velocities  and  accelerations 69 

Examples  X : 70 

XI. 

Successive  derivatives 73 

The  geometrical  meaning  of  the  second  derivative 73 

Points  of  inflexion 74 

Successive  differentials • 75 

Equicrescent  variables •  • 75 

Examples  XI •    • .   .  -    76 

CHAPTER    V. 
The  Evaluation  of  Indeterminate  Forms. 

XII. 

Indeterminate  or  illusory  forms 79 

Evaluation  by  differentiation 80 

Examples  involving  decomposition 82 

Examples  XII 84 

XIII. 

The  form  ^ 87 

Derivatives  of  functions  which  assume  an  infinite  value 89 

The  form  o  00   . . . . : 00 

The  form  00  —  00 no 

Examples  XIII ^. gi 

XIV. 


Functions  whose  logarithms  take  the  form  o  -oo  93 

The  form  i* g3 

The  form  0° 04 

Examples  XIV. g5 


CONTENTS.  VU 


CHAPTER    VI. 
Maxima  and  Minima  of  Functions  of  a  Single  Variable. 

XV. 

Conditions  indicating  the  existence  of  maxima  and  minima 97 

Maxima  and  minima  of  geometrical  magnitudes 99 

Examples  XV • loi 

XVI. 

Method  of  discriminating  between  maxima  and  minima 103 

Alternate  maxima  and  minima 104 

The  employment  of  a  substituted  function 106 

Examples  XVI 107 

XVII. 

Employment  of  derivatives  higher  than  the  first 109 

Complete  criterion  for  a  maximum  or  a  minimum ill 

Infinite  values  of  the  derivative 113 

Examples  XVII 114 

Miscellaiuous  examples  of  maxima  and  minima 115 

CHAPTER    VII. 

The  Development  of  Functions  in  Series. 

XVIII. 

The  nature  of  an  infinite  series I19 

Convergent  and  divergent  series 121 

Taylor's  theorem 122 

Lagrange's  expression  for  the  remainder 124 

The  binomial  theorem 126 

Examples  XVIII 127 

XIX. 

Maclaurin's  theorem 129 

The  exponential  series  and  the  value  of  /? 129 

Logarithmic  series 131 

Computation  of  Napierian  logarithms 132 

The  modulus  of  tabular  logarithms 134 

The  developments  of  sin  x  and  of  cos  x 134 

Examples  XIX 135 


Vlll  CONTENTS. 


CHAPTER  VIII. 
Curve  Tracing. 

XX.  PAGE 

Equations  in  the  form  y  —  f{x) '. 138 

Asymptotes  parallel  to  the  coordinate  axes 138 

Minimum  ordinates  and  points  of  inflexion 140 

Oblique  asymptotes 141 

Curvilinear  asymptotes 143 

Examples  XX 144 

XXI. 

Curves  given  by  polar  equations 146 

Asymptotes  determined  by  means  of  polar  equations 148 

Asymptotic  circles 149 

Examples  XXI 150 

XXII. 

The  parabola  of  the  nih.  degree  ...   151 

The  cubical  and  the  semicubical  parabolas 152 

The  cissoid  of  Diodes 153 

The  cardioid 154 

The  lemniscata  of  Bernoulli 154 

The  logarithmic  or  equiangular  spiral 155 

The  loxodromic  curve      155 

The  cycloid 157 

The  epicycloid 139 

The  hypocycloid 160 

The  four-cusped  hypocycloid 161 


CHAPTER    IX. 
Applications  of  the  Differential  Calculus  to  Plane  Curves. 

XXIII. 

The  equation  of  the  tangent  162 

The  equation  of  the  normal 163 

Subtangents  and  subnormals , 164 

The  perpendicular  from  the  origin  upon  a  tangent 165 

Examples  XXIII i66 


CONTENTS.  IX 


XXIV.  PAGE 

Polar  coordinates 167 

Polar  subtangents  and  subnormals 169 

The  perpendicular  from  the  pole  upon  a  tangent 170 

The  perpendicular  upon  an  asymptote I7I_ 

Points  of  inflexion 171 

Exaviples  XXIV 173 

XXV. 

Curv'ature 174 

The  direction  of  the  radius  of  curvature 176 

The  radius  of  curvature  in  rectangular  coordinates 177 

Expressions  for  p  in  which  x  is  not  the  independent  variable 178 

Examples  XXV 179 

XXVI. 

Envelopes 180 

Two  variable  parameters 183 

Evolutes 185 

Examples  XXVI 188 


CHAPTER   X. 
Functions  of  Two  or  More  Variables. 

XXVII. 

The  derivative  regarded  as  the  limit  of  a  ratio 190 

Partial  derivatives 191 

Examples  XXVII 194 

XXVIII. 

The  second  derivative  regarded  as  a  limit  194 

Higher  partial  derivatives ' 196 

Examples  XXVIII 198 


THE 

DIFFERENTIAL    CALCULUS 


CHAPTER   I. 
Functions,  Rates,  and  Derivatives. 


I. 

Functions, 

LA  quantity  whicn  depends  for  its  value  upon  another 
quantity  is  said  to  be  2i  function  of  the  latter  quantity.  Thus 
x",  tan;r,  \o^(a  4-  x),  and  a""  are  functions  of  x. 

The  quantity  upon  which  the  function  depends  must  be 
regarded  as  variable,  and  be  represented  in  the  analytical 
expression  for  the  function  by  an  algebraic  symbol.  This 
quantity  is  called  the  independent  variable.  It  is  essential 
that  variation  of  the  independent  variable  should  actually 
produce  variation  of  the  function.  Thus  the  quantities 
;tr'',  x"^  ^r{a  ■\-  x)  {a  —  x),  and  (tan  x  +  cot  ;ir)  sin  2x  are  not  func- 
tions of  X,  since  each  admits  of  expression  in  a  form  which 
does  not  involve  x, 

2.  The  notation  /{x)  is  employed  to  denote  any  function 
of  X,  and,  when  several  functions  of  x  occur  in  the  same  in- 


2  FUNCTIONS  RA  TES  AND  DERIVA  TIVES.  [Art.  2. 

vestigation,  such  expressions  as  F{x\  F'  {x),  (^  (;r),  etc.,  are 
employed,  the  enclosed  letter  always  denoting-  the  indepen- 
dent variable.  When  expressions  like  f{}),f{a\  f{2x),  or 
/(o)  are  employed,  it  must  be  understood  that  the  enclosed 
quantity  is  to  be  substituted  for  x  in  the  expression  which 
defines  f(x).     Thus,  if  we  have 

f{x)  =  x'  +  X, 
/(i)  =z  2,        /{2x)  =  4x'  +  2x,        and       /(o)  =  c. 

Again,  if  F{x)  =  log„;t:  {a>  \) 

F{i)  =  o,  F{6)  =  —  00,  and         F{a)  =  i. 

3.  When  x  denotes  the  independent  variable  upon  which 
a  function  depends,  any  quantity  independent  of  x  is,  in  con- 
tradistinction, called  a  constant ;  both  when  it  is  an  absolute 
constant,  like  i,  |/2,  or  tt,  and  when  it  is  denoted  by  a  symbol, 
like  a,  ?/,  or  y^  to  which  any  value  can  be  assigned.  Thus, 
when  a'  is  denoted  by  /(^),  it  is  considered  simply  as  a  func- 
tion of  X,  and  a  is  regarded  as  a  constant. 

When  it  is  desired  to  express  that  a  quantity  is  a  function 
of  two  quantities,  both  the  symbols  denoting  them  are  placed 
between  marks  of  parenthesis.  Thus,  since  a^  is  a  function  of 
X  and  a,  we  may  write 

fixy  d)  =  a^. 
Accordingly  we  have 

Ay.b)  =  b\         /(3,  2)  =  8,         and         /(2,  3)  =  9-    • 

4.  It  is  often  convenient  to  represent  the  value  of  a  func- 
tion of  X  by  a  single  letter  ;  thus,  for  example,  y  =  x^.  When 
this  notation  is  used,  if  we  represent  the  independent  variable 
X  by  the  abscissa  of  a  point,  and  the  function  y  by  the  corre- 


§   I.]  IMPLICIT  FUNCTIONS,  3 

spending  ordinate,  a  curve  may  be  constructed  which  will 
graphically  represent  the  function,  and  will  serve  to  illustrate 
its  peculiarities. 

Rectangular  coordinates  are  usually  employed  for  this 
purpose.     See  diagram,  Art.  lo. 

A  function  of  the  form 

y  z=^  mx  -\-  b^ 

m  and  b  being  constants,  is  represented  by  a  straight  line. 
Functions  of  this  form  are,  for  this  reason,  called  linear  func- 
tions, 

l77tplicit  Functions. 

5.  When  an  equation  is  given  involving  two  variables  x 
and  J/,  either  variable  is  obviously  a  function  of  the  other ; 
and  the  former  variable,  when  its  value  is  not  directly  ex- 
pressed in  terms  of  the  other,  is  said  to  be  an  implicit  func- 
tion of  the  latter.     Thus,  if  we  have 

ax"^  —  "^axy  +  y'  —  ^'  =  o, 

either  variable  is  an  implicit  function  of  the  other. 
By  solving  the  above  equation  for  x,  we  obtain 


/(--f-fl- 


,=f±/(.'.f 


In  this  form  of  the  equation,  x  is  said  to  be  an  explicit  func- 
tion of  y. 

This  example  will  serve  to  illustrate  the  fact,  that  from  a 
single  equation  involving  two  variables,  there  may  be  derived 
two  or  more  explicit  functions  of  the  same  variable.  In  the 
above  case,  x  is  said  to  be  a  two-valued  function  of  y ;  while, 
since  the  equation  is  of  the  third  degree  in  y,  the  latter  is  a 
three-valued  function  of  x. 


FUNCTIONS  RA  TES  AND  DERIVA  TIVES.  [Art.   6. 


Inverse  Functions. 

6.  \i  y  =  f{^)y  ■*■  is  some  function  oi  y  ;  we  may  therefore 
write 

y—f{x),  whence  x  ±^  (t>{y). 

Each  of  the  functions /and  </>  is  then  said  to  be  the  inverse 
function  of  the  other.     Thus,  if 

y  =  a"^,  we  have  x  =  logay  ; 

hence  each  of  these  functions  is  the  inverse  of  the  other.    So 
also  the  square  and  the  square  root  are  inverse  functions. 

7.  In  the  case  of  the  trigonometric  functions,  a  peculiar 
notation  for  the  inverse  functions  has  been  adopted.  Thus, 
if  we  have 

X  =  sin  6,  we  write  6  =  sin  ~^x. 

Whenever  trigonometric  functions  are  employed  in  the 
Calculus,  the  symbol  representing  the  angle  always  denotes 
the  circu/ar  measure  of  the  angle  ;  that  is,  the  ratio  of  the  arc 
to  the  radius.  Hence  sin~*;ir  maybe  read  either  "the  in- 
verse sine  of  x''  or  "  the  arc  whose  sine  is  ;r." 

The  inverse  trigonometric  functions  are  evidently  many- 
valued.     See  Art.  54. 

The  Classification  of  Functions, 

8.  With  reference  to  its  formy  an  explicit  function  is 
either  algebraic  or  transcendental. 

An  algebraic  function  is  expressed  by  a  definite  combination 
of  algebraic  symbols,  in  which  the  exponents  do  not  involve 
the  independent  variable. 


§    I.]  THE  CLASSIFICATION  OF  FUNCTIONS.  5 

All  functions  not  algebraic  are  classed  as  transcendental. 
Under  this  head  are  included  exponential  functions ;  that  is, 
those  in  which  one  or  more  exponents  are  functions  of  the 
variable,  as,  for  example,  a^,  xa^^,  etc. :  logarithmic  func- 
tions :  the  direct  and  inverse  trigonometric  functions,  and 
other  forms  which  arise  in  the  higher  branches  of  mathematics. 

9.  With  reference  to  its  mode  of  variation,  a  function  is 
said  to  be  an  increasing  fimction  when  it  increases  and  de- 
creases with  X ;  and  a  decreasing  function  when  it  decreases 
as  X  increases,  and  increases  as  x  decreases.  Thus,  it  is  evi- 
dent that  x^  is  always  an  increasing  function  of  x,  while  —  is 

always  a  decreasing  function  of  x.  Again,  tan  x  is  always  an 
increasing  function,  but  sin;ir  is  sometimes  an  increasing  and 
sometimes  a  decreasing  function  of  x, 

10.  The  increase  and  decrease  here  considered  are  algc- 
hraic.  For  example,  x"^  is  an  increasing  function  when  x  is 
positive,  but  when  x  is  negative  it  becomes 
a  decreasing  function  ;  for,  when  x  is  negative 
and  algebraically  increasing,  x"^  is  decreasing. 

The  curve  y  ^=  x"^  which  illustrates  this 
function  is  constructed  in  Fig.  i.  Since  alge- 
braic increase  in  the  value  of  x  is  represented 
by  motion  from  left  to  right,  whether  the 
moving  point  is  on  the  left  or  on  the  right  of 
the  axis  of  y,  the  downward  slope  of  the  curve  on  the  left 
of  the  origin  indicates  that  x""  is  a  decreasing  function  when  x 
is  negative. 

Expressions  involving  an    Unknown  Function. 

11.  An  expression  involving  f{x\  as,  for  example,  xf{x) 
or  F\^f{x)\,  is  generally  a  function  of  x\  but  it  may  happen 


6  FUNCTIONS  RA  TES  AND  DERIVA  TIVES.        [Art.  1 1. 

that  such  an  expression  has  a  value  independent  of  x.  Thus, 
suppose  that,  in  the  course  of  an  investigation,  the  following 
equation  presents  itself : — 

xf{x)  =  zf{z\ 

in  which/  denotes  an  unknown  function,  and  x  and  z  are  en- 
tirely independent  arbitrary  quantities.  When  this  is  the  case, 
we  can  make  z  a  fixed  quantity,  and  give  to  x  any  value  what- 
ever; that  is,  we  can  make  x  a  variable  and  z  a  constant; 
but  if  z  is  a  constant,  zf(z)  is  likewise  a  constant,  v/e  can, 
therefore,  write 

xf{x)  —  c, 

c  being  an  unknown  constant.     Hence  we  have 

The  value  of  the  constant  c  is  readily  found,  if  we  know  the 
value  of  f{x)  corresponding  to  any  one  value  of  x. 


Examples  I. 

1.  {pc)  For  what  value  of  n  does  x^  cease  to  be  a  function  of  x} 
(0)  For  what  values  of  x  does  it  cease  to  be  a  function  of  n? 

(a)  When  n  =  o.     (/3)  When  ^  =  i,  or  ;ir  =  o. 

2.  Ifyfi '- — -^j=,r+  — ^—,  show  that  V  is  a  function  of  «,  but 

-^  \         a  +  xj  a  +  X 

not  of  X. 

3.  Show  that  sin,r  tan  ^x  +  cos;r  is  not  a  function  of  x. 

4.  U  y  =  X  +  4/(1  +  X-),  show  that/"  —  2xjy  is  not  a  function  of  x, 

5.  If /(^)  =  x\  find  the   value  o( /{x  +  Ji)\  of/(2;r);    of /(^'')  ;  of 
f{x^-x)',  of/(i);/(i2);/[/(^)]. 

fix  +  y^)  =  ;ir'  +  2/^  ;r  +  h^. 


§   I.]  EXAMPLES  OF  FUNCTIONS.  7 

6.  If /(^)  =  COS0,  find  the  value  of /(o)  ;    of  /(^tt)  ;    o\  /(i^r)  ;    of 

7.  If  F{x)  —  ax,  give  the  value  of  F{d)\  of  F{\)\  of  /^(o).     Also 
show  that  in  this  case  \jF{x)Y  —  F  {7.x). 

8.  Given  j^  —  7.ay  +  ;ir-  =  o,  make_y  an  explicit  function  of  x. 

y  —  a±  ^{0"  —  X-). 

9.  Given  i  +  loga  _y  =  2  log^  {x  +  «),  make ^y  an  explicit  function  of  x. 

{x-^ay 

10.  Given  the  equations — 

71  -\-  \  —71  (cos^^'  +  cos  Q'  cos  0  +  cos'^^, 
and  n  —  I  —  n  (sin'^^'  +  sin  6'  sin  6  +  sin^^)  ; 

eliminate  n,  and  make  ^''  an  explicit  function  of  6.    Also  make  n  an  ex- 
plicit function  of  ^.  ,  i 

e'  =zd±^Tz,  and  ;2  =  T 


sin  <^  cosO 


11.  Given  sin  —  '  jr  +  sin'" ' y  =  a,  makej/  an  explicit  function  of  x. 

y  =  sin  a  4/(1  —  x^)  —  x  cos  or. 

12.  Given  tan-';ir  +  tan~'/  =  <ar,  make^  an  explicit  function  of  jr. 

tan  a—  X 
y  ~  I +^  tana' 

13.  Given  xy  —  2x  +y  =  n,  show  that  j  is  not  a  function  of  x  when 
n  =  2. 

2X  —  I 

14.  l(  y  =  - ,  show  that  the  inverse  function  is  of   the  same 

jX  —  2 

form. 

1  +  X 

15.  Ity  —f{x)  =  —3--,  find  z  =f{^y),  and  express  ^  as  a  function 

of  X.  I 

~      ^' 

16.  If  both  /  and  (p  denote  increasing  functions,  or,  if  both  denote 
decreasing  functions,  show  that  ^[/{x)]  is  an  increasing  function. 
Also  show  that  the  inverse  of  an  increasing  function  is  an  increasing 
function. 


8     '  FUNCTIONS  RA  TES  AND  DERIVA  TIVES.  [Ex.  I. 

17.  Find  the  inverse  of  the  function,  j  =  log^  [x  +  /^(i  +  x")]. 

.   x=  ^{ev-e-y). 

18.  lif{x)  be  an  unknown  function  having  the  property 

prove  that  /{i)  =  o. 

Futy  =  I. 

19.  If /(^)  has  the  property 

f{x+y)=/{x)+f{y), 
prove  that/(o)  =  o.    Also  prove  that  the  function  has  the  property 

f{Px)=p/{x), 

in  which/  is  a  positive  or  negative  integer. 

For  positive  integers,  put  y  =  x,  ix,  ^x,  etc.^  in  the  given  equation  ;  for 
negative  integers,  put  y  =  —  x. 

20.  If /denotes  the  same  function  as  in  Example  19,  prove  that 

/i?nx)  =  m/{x), 

m  denoting  any  fraction. 

Solution : — 

^      .  P 

Puttmg  z  =  —x^  qz  —px, 

f{^^)=f{p^)\ 
hence,  by  Example  19,  q/{^)  =Pf  {x)> 

or  /(^)=^/(-^), 


/(H=,^/«. 


21.  Given,  the  property  of  the  same  function  proved  in  Example  20; 
f{mx)  =  mf{x)\ 


§  I.]  EXAMPLES  OE  EUNCTIONS. 

by  putting  2  for  mx,  show  that 

and  thence  deduce  the  form  of  the  function.     See  Art.  11. 

/(^)  =  ex. 
22.  Given,  [^  {x)Y  =  [?>  (2')>  ,  and  9  (i)  =  s,  ' 


determine  <}>  {x). 

23.  Given  0  (^)  +  ^  ( j)  =  <j>  {xy) 

prove  ^  {xf")  =  m  <!>  (x), 

and  thence  prove  (p  (x)  =  c  logo:. 

Use  the  7)iethods  of  Examples  19,  20,  and  21. 


9(^)  =  e^ 


II. 

Rates, 

12.  In  the  Differential  Calculus,  variable  quantities  are 
regarded  as  undergoing  continuous  variation  in  magnitude, 
and  the  rates  of  variation,  denoted  by  appropriate  symbols, 
are  employed  in  connection  with  the  values  of  the  variables 
themselves. 

If  a  varying  quantity  be  represented  by  the  distance  of  a 
point  moving  in  a  straight  line  from  a  fixed  origin  taken  on 
that  line,  the  velocity  of  the  moving  point  will  represent  the 
rate  of  increase  or  decrease  of  the  varying  quantity. 


Fig.  2. 


Thus  O  (Fig.  2)  being  the  fixed  origin  and  (9/*  a  variable 
denoted  by  x,  P  is  the  moving  point  whose  velocity  repre- 
sents the  rate  of  x.  The  velocity  of  P,  or  the  rate  of  x,  is 
regarded  as  positive  when  P  moves  in  the  direction  in  which 
X  increases  algebraically  ;  thus,  taking  the  direction  OX^  or 
toward  the  right,  as  the  positive  direction  in  laying  off  x,  the 


lO  FUNCTIONS  RATES  AND  DERIVATIVES.        [Art.  12. 

velocity  is  positive  when  P  moves  toward  the  right,  whether 
its  position  be  on  the  right  or  on  the  left  of  the  origin.  Ac- 
cordingly, a  rate  of  algebraic  decrease  is  considered  as  nega- 
tive, and  would  be  represented  by  a  point  moving  toward  the 
left. 

Constant  Rates, 

(3.  The  rate  of  a  quantity  like  the  velocity  of  a  point  may 
be  either  constant  or  variable.  A  velocity  is  uniform  or  con- 
stant, when  the  spaces  passed  over  in  any  equal  intervals  of 
time  are  equal,  or,  in  other  words,  wJie7i  the  spaces  passed  oveis* 
in  any  intervals  of  time  are  proportiojial  to  the  intervals. 

The  numerical  measure  of  a  uniform  velocity  is  the  space 
passed  over  in  a  unit  of  time  ;  then  if  t  denote  the  time  elapsed 
from  an  assumed  origin  of  time,  and  k  the  space  passed  over 
by  a  moving  point  in  a  unit  of  time,  kt  will  denote  the  space 
passed  over  in  the  time  /.  Hence,  whenever  the  velocity  is 
uniform,  the  quotient  obtained  by  dividing  the  number  of 
units  of  space  by  the  number  of  units  of  time  occupied  in 
describing  this  space  is  constant,  and  serves  as  the  numerical 
measure  of  the  velocity. 

14.  Now,  if  ;ir  be  a  quantity  having  a  uniform  rate  k,  it 
will  be  represented  by  the  distance  from  the  origin  of  a  point 
having  the  uniform  velocity  k,  and  if  a  denote  the  value  of  x 
when  /  is  zero,  we  shall  have 

X  =^  a  +  kt (i) 

This  formula  expresses  a  uniformly  varying  quantity  as  a 
function  of  /.  When  ;r  is  a  uniformly  decreasing  quantity, 
k  is,  of  course,  negative. 

Conversely,  if  x,  when  expressed  as  a  function  of  /,  is  of  the 
form  (i),  involving  the  first  power  only  of  /,  then  ;r  is  a  quan- 
tity having  a  uniform  rate,  and  the  coefficient  ^  is  a  measure 
of  this  rate. 


§  n.]  VARIABLE   VELOCITIES.  II 

Variable    Velocities, 

15.  If  the  velocity  of  a  point  be  not  uniform,  its  numerical 
measure  at  any  instant  is  the  number  of  units  of  space  which 
would  be  described  in  a  unit  of  time^  were  the  velocity  to  remain 
constant  from  and  after  the  given  instaiit. 

Thus,  when  we  speak  of  a  body  as  having  at  a  given  in- 
stant a  velocity  of  32  feet  per  second,  we  mean  that  should  the 
body  continue  to  move  during  the  whole  of  the  next  second, 
with  the  same  velocity  which  it  had  at  the  given  instant,  32 
feet  would  be  described.  The  actual  space  described  may  be 
greater  or  less,  in  consequence  of  the  change  in  velocity  which 
takes  place  during  the  second  ;  it  is,  for  instance,  greater  than 
the  measure  of  the  velocity  at  the  beginning  of  the  second, 
in  the  case  of  a  falling  body,  because  the  velocity  increases 
throughout  the  second. 

16.  Attwood's  machine  for  determining  experimentally 
the  velocities  acquired  by  falling  bodies  furnishes  a  familiar 
example  of  the  practical  application  of  the  principle  em- 
bodied in  the  above  definition. 

This  apparatus  consists  essentially  of  a  thread  passing 
over  a  fixed  pulley,  and  sustaining  equal  weights  at  each  ex- 
tremity, the  pulley  being  so  constructed  as  to  offer  but  slight 
resistance  to  turning.  On  one  of  the  weights  a  small  bar  of 
metal  is  placed,  which,  destroying  the  equilibrium,  causes  the 
weight  to  descend  with  an  increasing  velocity.  To  deter- 
mine the  value  of  this  velocity  at  any  point,  a  ring  is  so  placed 
as  to  intercept  the  bar  at  that  point,  and  allow  the  weight  to 
pass.  Thus,  the  sole  cause  of  the  variation  of  the  velocity 
having  been  removed,  the  weight  moves  on  uniformly  with 
the  required  velocity,  and  the  space  described  during  the 
next  second  becomes  the  measure  of  this  velocity. 


12  INUNCTION'S  RA  TES  AND  DERIVA  TIVES.        [Art.  I/. 

Variable  Rates. 

17.  When  ;r  is  a  function  of  /,  but  not  of  the  form  ex- 
pressed by  equation  (i),  Art.  14— that  is,  when  the  function  is 
not  linear — the  rate  of  x  will  be  variable.'  To  obtain  the 
measure  of  this  rate  at  any  given  instant,  we  employ  the 
same  principle  as  in  the  case  of  a  variable  velocity.  Thus, 
let  X  be  represented  by  O P\s>  in  Fig.  2,  Art.  12,  let  the  sym- 
bol dt  denote  an  assumed  interval  of  time,  and  let  dx  denote 
the  space  which  would  be  described  in  the  time  dt,  were  P 
to  move  with  the  velocity  which  it  has  at  the  given  instant 
unchanged  throughout  the  interval  of  time  dt.  Then  the 
space  which  would  be  described  in  a  unit  of  time  is,  evidently, 

dx 
.      'dt' 

which  is  therefore  the  measure  of  the  velocity  of  /*,  or  the 
rate  of  x. 

This  ratio  is  in  general  variable,  but,  when  x  is  of  the  form 
a-\-  kt/\\,  has  been  shown  in  Art.  14  that  k  is  the  measure  of 
the  rate ;  we  therefore  have 

=  k,  when  X  =  a  -{-  kt. 


dt 


Differ  en  tials. 


(8.  The  quantities  dx  and  dt  are  called  respectively  "the 
differential  of  ;r"  and  "the  differential  of  /." 

In  accordance  with  the  definition  of  dx  given  in  the  pre- 
ceding article,  the  differential  of  a  variable  quantity  at  any 
instant  is  the  increment  which  would  be  received  in  the  time 
dt,  were  the  quantity  to  continue  to  increase  uniformly 
during  that  interval  of  time  with  the  rate  it  has  at  the  given 


§  II.]  DIFFERENTIALS.  13 

instant.  The  quotient  obtained  by  dividing  the  differential  of  any 
quantity  by  dt  is  tJierefore  the  measure  of  the  rate  of  the  quantity. 
The  differential  of  a  quantity  is  denoted  by  prefixing  d  to 
the  symbol  denoting  the  quantity  ;  when  the  symbol  denot- 
ing the  quantity  is  not  a  single  letter  it  is  usually  enclosed 
by  marks  of  parenthesis  to  avoid  ambiguity.  Thus,  d{x'')y 
d(xy),  ^(tan;r),  d{a^  +  x^),  etc. 

T/ie  Differentials  of  Polynomials, 

19.  Let  X  and  y  denote  two  variable  quantities,  and  let  a 
and  b  denote  particular  simultaneous  values  of  x  and  y^  while 
k  and  k'  denote  corresponding  values  of  the  rates  of  x  and  y. 

Now,  if  X  and  y  should  continue  to  vary  with  these  rates, 
their  values  would  (see  Art.  14)  be  expressed  by 


x  =  a^kt, 

and 

y=b  +  k't, 

whence 

x+y  =  a-\-b+{k^  k')t. 

Thus  the  quantity  ;r+ J  would  become  a  uniformly  varying 
quantity,  and,  by  Art.  14,  its  rate  would  be  k-\-k\  which, 
therefore,  is  the  measure  of  the  rate  of  x  -\-y  at  the  instant 
when  X  and  y  have  the  rates  k  and  k\     Consequently, 

dt     -^+^  -  dt^ dt' 

Now,  since  k  and  k'  denote  any  values  of  the  rates,  this  equa- 
tion is  universally  triie.     We  have,  therefore, 

d{x^y)  =  dx^dy (l) 

This  formula  is  easily  extended  to  the  sum  of  any  number 
of  variables.     Thus, 

dix^y  \z^r'  ")^  dx\d{^y\2-\-  *  *  ^)=^  dx-^-dy-irds-^^ (2) 


14  FUNCTIONS  RA  TES  AND  DERIVA  TIVES.  [Art.  20. 

20.  The  differential  of  a  constant  is  evidently  zero,  hence 

d{x-\-h)  =  dx (3) 

Again,  if  y  z=  —  x,  y-\-x  =  Oy 

hence,  by  equation  (i),  since  zero  is  a  constant,  we  have 

dy  -\-  dx  =^  o,  or  dy  =^  —  dx  ; 

that  is,  d,{—x)  =  —  dx (4) 

The  differential  of  a  negative  term  is  therefore  the  negative 
of  the  differential  of  the  term  taken  positively. 

It  appears,  on  combining  the  results  expressed  in  equations 
(2),  (3),  and  (4),  that  t/ie  differential  of  a  polynomial  is  the  alge- 
braic sum  of  the  differentials  of  its  terms;  and  that  constant 
terms  disappear  from  the  result. 

The  Differential  of  a   Term  having  a  Constant 
Coefficient. 

21.  Let  the  term  be  denoted  by  mxy  m  denoting  a  con- 
stant. 

Resuming  equation  (2),  Art.  19 ;    viz., 

d{x -vy^  z^-  •  •  •)  =  dx ■\- dy -^  dz -\-  • .  •, 
and  denoting  the  number  of  terms  by  /,  we  put 

x=y—z— , 

thus  obtaining  d{px)=pdx, (i) 

p  denoting  an  integer. 


§  II.]  THE  DIFFERENTIAL  OF^mx).  1 5 


To  extend  equation  (i)  to  the  case 
fraction,  let 

in  which  m  denotes  a 

s  =  -  X,                then 

qz=px. 

By  applying  equation  (i)  we  obtain 

qd2=pdxj                or 

d.  =  ^dx; 

that  is,                                 ^[-A  =  ~^^' 

Hence  generally,  when  m  is  positive, 

d{in  x)  —  in  dx (2) 

Since  d{—  x)  =  —  dx,  this  equation  is  true  likewise  when  m 
is  negative. 

It  therefore  follows  that  t/ie  differential  of  a  term  having  a 
constant  coefficient  is  equal  to  the  product  of  the  differential  of  the 
variable  factor  by  the  constant  coefficient. 

Examples  XL 

I.  Find  the  differential  of   — ,  and  of 


3«'  w  —  2  *  2dx        ,     dx 

,  and 

3<3:  m  —  2 


2.  Find  the  differential  of 1^-  ,  and  of  ^- 


dx        ,     dx 
— T,  and 2' 


^     ,  ,.«•  .  ,     ,    a  +  d  +  (a  —  b)x  dx 

3.  Find  the  differential  of 7, -r —ri' 

4.  Find  the  differential  of —7,  and  of  —, — -^  . 

^  a  -{■  0  a{a  +  o) 

dx  J    b{dx  +  dy) 

a  +  b'  a{a+b) 


l6  FUNCTIONS  RATES  AND  DERIVATIVES.  [Ex.  II. 

dv 

5.  Given  ay  ■\-  bx  ■\-  2cx  ■\-  ab  =  o,\.q  find  -j-.  ,  , 

■'  ^   '  '  ^x  dy  _       b  +  2c 

dx  ~  a      * 

dy 

6.  Given  y  log  a  -^  x  sin  a  —y  cos  a  —  « jir  +  tan  a  =  o,  to  find  ~. 

dy'  a  —  sin  a 


7.  Given  ay  cos'* a  ~2b{i  —  s\vi(x)x  —  b{a  —  x  cos^ a), to  find ^ 


dx        log  rt  —  cos  oc  ' 

s^  a),  to  find  ^. 

</k      <^  (i  —  sin  a) 
dx~  a  (i  +  sin  a)' 


8.  Given  ^'^  +  2  (i  +  cos  a)y  =  {x  +  y)  sin'^  a,  to  find  —-- 

dy  ^  ^a 
~r  =  tan^  — 
dx  2 

X      y      " 

9.  Given  — h -r  +  —  =  i,  to  express  ds  in  terms  of  ^^  and  dy. 

dz  =z dx  —  7-  dy. 

a  b    -^ 

10.  A  man  whose  height  is  6  feet  walks  directly  away  from  a  lamp- 
post at  the  rate  of  3  miles  an  hour.  At  what  rate  is  the  extremity 
of  his  shadow  travelling,  supposing  the  light  to  be  10  feet  above  the 
level  pavement  on  which  he  is  walking? 

Draw  a  figure.,  a7id  denote  the  variable  distance  of  the  man  fro7n  the 
lamp-post  by  x,  and  the  distance  of  the  extremity  of  his  shadow  from  the 
post  by y.  7i  miles  per  hour. 

11.  At  what  rate  does  the  man's  shadow  (Ex.  10)  increase  in  length  ? 


III. 

Differentials  of  Functions  of  aft  Independent   Va7'iable, 

22«  When  the  variables  involved    in    any  mathematical 

investigation  are  functions  of  an  independent  variable  x^  the 

dx 
latter  may  be  assumed  to  have  a  rate  denoted  by  -— ,  in  which 


§^  III.]  THE  DERIVATIVE.  1/ 

dx  is  arbitrary.  So  also  the  corresponding  rate  of  y  will  be 
denoted  by  — ,  and,  if  jj/  is  a  function  of  x,  the  value  of  dy  will 

depend  in  part  upon  the  assumed  value  of  dx. 

To  differentiate  a  function  of  x  is  to  express  its  differential 
in  terms  of  x  and  dx. 

It  is  to  be  understood,  of  course,  that  the  differentials 
involved  in  an  equation  are  all  taken  with  reference  to  the 
same  value  of  dt. 

If  two  quantities  are  always  equal,  their  simultaneous 
rates  are  evidently  equal;  and  hence  their  dift'erentials  are 
likewise  equal.  We  can  therefore  differentiate  an  equation  ; 
that  is,  express  the  equality  of  the  differentials  of  its  mem- 
bers; provided  the  equation  is  true  for  all  values  of  the 
variables  involved.     Thus,  from  the  identical  equation 

{x-^h)^=.x''^2hx^h\ 

it  follows  that       d\{x  +  Jif'\  =  d{x')  +  2/1  dx. 

The   Derivative. 

23-  Before  proceeding  to  the  differentiation  of  the  vari- 
ous functions  of  x,  it  is  necessary  to  show  that,  it 

^=/W. (I) 

the  ratio  -f- 

dx 

has  a  definite  value  for  each  value  of  x,  independent  of  the  assumed 
value  ^/dx. 

Let  a  particular  value  of  x  be  denoted  by  a,  and  let  the 
corresponding  value  of  dx  be  an  arbitrary  quantity. 

Now,  although  dx  is  arbitrary,  since  dt  is  likewise 
arbitrary,  the  rate  of  x^  that  is,  the  ratio 

§ (^) 


1 8  FUNCTIONS  RATES  AND  DERIVATIVES.       [Art.  23. 

may  be  assumed  to  have  a  certain  fixed  value  at  the  instant 
when  X  =^  a.  The  corresponding  value  of  the  rate  of  y, 
denoted  by 

dt' ^^^ 

» 

evidently  depends  solely  upon  the  rate  of  x  and  upon  the  form 
of  the  function /in  equation  (i).  Hence,  when  the  value  of 
the  rate  (2)  is  fixed,  the  value  of  (3)  is  also  definitely  fixed. 

Denoting  these  fixed  values  by  k  and  k' ,  we  have,  when 
;r  =  ^, 

dx       J         A  dy       J,       .  dy       k' 

—-  z=.  k,  and  -^  :=  k ,  whence  -^  =^  -- 
dt         '  dt  '  dx      k 

Hence,  corresponding  to   a   particular  value  a  of  x,  there 

exists  a  determinate  value  -»of  the  ratio  -f-»  notwithstand- 

k  dx 

ing  the  fact  that  dx  has  an  arbitrary  value;  in  other  words, 

the  value  of  the  ratio  — ^  is  independent  of  the  arbitrary  value  ^/dx. 
dx 


24-  It  is  obvious  that,  in  general,  this  ratio  will  have 
different  values  corresponding  to  different  values  of  x,  and 
hence  that  it  may  be  expressed  as  a  function  of  x^  and  de- 
noted by/X^) ;  thus, — • 

£=^'(-) (') 

The  form  of  this  new  function/'  will  evidently  depend  upon 
that  of  the  given  function  / 

The  function /'(;r)  is  called  the  derivative  of /(;r),  and,  since 
equation  (i)  may  be  written  in  the  form 

dy  =  f'(pc)  dxy 

it  is  also  called  the  differential  coefficient  of  y  regarded  as  a 
function  of  x. 


§   III.]      GEOMETRICAL  MEANING  OF  THE  DERIVATIVE.         I9 

When,  however,  the  given  function  f{x)  is  of  the  hnear 

form 

jj/  =  mx  +  b, 

the  derivative  is  no  longer  a  function  of  Xy  but  is  a  constant, 
since  the  value  of  y  gives 

dy  =^  m  dxy 

dy 
or  -^^  =  m. 


The    Geometrical  Meaning   of  the  Derivative, 

25-  Representing  the  corresponding  values  of  x  and/  by 
the  rectangular  coordinates  of  a  moving  point,  if  this  point 
move  in  a  uniform  direction,  so  as  to  describe  a  straight  line, — 

tliat  is,  if  J/  be  a  linear  function  of  Xy — the  value  of  —r   will  be 

constant,  by  the  preceding  article.  Hence,  in  the  general 
case,  when  this  ratio  is  variable,  the  point  will  move  in  a  vari- 
able direction. 

If  we  denote  the  inclination  of  this  direction  to  the  axis  of 
X  by  <A,  the  value  of  </>  will  vary  with  the  value  of  Xy  and  the 
point  will  describe  a  curve. 

The  tangent  line  to  a  curve  is  defined  as  follows : — 

The  tangent  to  a  curve  at  any  point  is  the  straight  line  which 
passes  through  the  pointy  and  has  the  direction  of  the  curve  at  that 
point!^ 

Hence,  for  any  point  of  the  curve,  0  denotes  the  inclina- 
tion to  the  axis  of  x  of  the  tangent  line  at  that  point. 


*  It  will  be  shown  hereafter  (Art.  49)  that,  in  the  case  of  the  circle,  this 
general  definition  of  a  tangent  line  agrees  with  that  usually  given  in  Plane 
Geometry. 


20  FUNCTIONS  RA  TES  AND  DERI V A  TIVES.       [Art.  26. 

26.  Now,  if  a  point,  at  first  moving  in  the  curve,  should, 
after  passing  the  point  whose  abscissa  is  a,  so  move  that  the 

rates  -7-  and  —r  retain  the  values  which  they  had  at  the  in- 
at  at 

stant  of  passing  the  given  point,  the  direction  of  its  motion 
will  become  constant,  and  the  point  will  describe  a  straight 
line  tangent  to  the  curve  at  the  given  point. 

The  value  of  dx  may  be  repre- 
sented by  an  arbitrary  increment  of 
X  as  in  Fig.  3  ;  the  value  of  dy  will 
then  be  represented  by  the  corre- 
sponding increment  which  would  be 
received  by  y,  were  the  point  moving 
in  the  tangent   line,  as   indicated   in 


Fig.  3. 


the  diagram.     Hence 


dy 

-y-  =  tan  0, 
ax 


which  is  evidently  independent  of  the  assumed  value  oidx.^ 
It  follows  that  the  value  of  the  derivative  of  f(x),  for  any 
value  of  X,  is  represented  by  the  trigonometric  tangent  of 
the  inclination  to  the  axis  of  x  of  the  curve  y  =/(;ir),  at  the 
point  corresponding  to  the  given  value  of  x, 

27-  The  moving  point,  which  is  conceived  to  describe 
the  curve,  may  pass  over  it  in  either  of  two  directions  differ- 
ing by  180°.  The  two  corresponding  values  of  0  give,  how- 
ever, the  same  value  of  tan  <^,  since  tan  (^  ±  180°)  =  tan  ^. 

Thus,  in  Fig.  3,  the  point  P  may  be  regarded  as  moving 
so  as  to  increase  x  and  7,  in  which  case  both  dx  and  dy  will 
be  positive,  and   ^  will  be  in   the  first  quadrant ;  or  P  may 


*  In  other  words,  the  value  of  the  derivative  is  determined  by  the  form  of  the  function/"  which 
determines  the  curve,  and  the  value  ol  x  which  fixes  the  position  of/*. 


§   III.]     GEOMETRICAL  MEANING  OF  THE  DERIVATIVE.  21 

move  in  the  opposite  direction,  making  dx  and  d^  negative, 
and  placing  0  in  the  third  quadrant.  In  either  case,  ~  or 
tan(/)  is  positive. 

28.  It  is  evident  that  when  f{x)  is  an  increasing  func- 
tion, as  in  Fig.  3,  ~j~  is  positive,  and  that  when  it  is  a  de- 

.       .         .         dy  . 
creasing  function,  -r-  is  negative. 

Thus  the  sign  of  f  {x)  for  any  value  of  x  is  positive  or 
negative  according  as  f(x)  is,  for  that  value  ot  x,  an  increas- 
ing or  a  decreasing  function.  For  example,  it  is  evident  that 
the  value  of  the  derivative  of  sin;ir  must  be  positive  when  x 
is  between  o  and  ^tt,  negative  when  x  is  between  ^tt  and  |- t, 
and  so  on. 

When  the  notation  —r-  is  used,  the  value  of  the  derivative 
dx 

corresponding  to  a  particular  value  ^  of  ;ir  is  expressed  by 

dy~\ 

~      which  is  equivalent  to/' (^).     See  Art.  2. 

Examples  III. 

1.  If  a  point  move  in  the  straight  line  2y  —  7.r  —  5  =  o,  so  that  fts 
ordinate  decreases  at  the  rate  of  3  units  per  second,  at  what  rate  is  the 
point  moving  in  the  direction  of  the  axis  of  ^? 

dx^_6_ 
dt  7' 

2.  If  a  point  starting  from  (o,  b)  move  so  that  the  rates  of  its  co- 
ordinates are   k  and  k',  show  that  its  path  \s  y=  jnx  -\-  b,   7n   being 

k' 
equal  to  7- 

Express  x  and  y  in  terms  of  t  {Art.  14),  and  eliminate  t. 

3.  If  a  point  moving  in  a  curve  passes    through  the  point  (5,  3) 


22  FUNCTIONS  RATES  AND  DERIVATIVES.        [Ex.  III. 

moving  at  equal   rates  upward  and  toward  the  left,  find  the  value  of 
— ^     ,  also  the  equation  of  the  tangent  line  to  the  curve  at  the  given 

point.  -^-      =  —  I,  and^  +  ;f  =  8. 

4.  If  a  point  is  moving  in  the  straight  line 

;r  cos  a  +  J  sin  a  = /, 

its  rate  in  the  positive  direction  of  the  axis  of  x  being  /  sin  ex.,  what  is  its 

rate  of  motion  in  the  direction  of  the  axis  of  ^? 

—  /  cos  oc, 

5.  Given  ay  sin«  —  a;ir+ajr  cosa  —  ^^  sec  a  =  o;  show  that  0  is  con- 
stant and  equal  to  \a. 

6.  U/{x)  =  tan;ir,  show  that/'(;r)  must  always  be  positive. 

7.  Show,  by  trac'ng  the   curve,   that  if  j  =  x^,  _Z  can  never  be 
negative. 


CHAPTER   11. 

The  Differentiation  of  Algebraic  Functions. 


IV. 

The  Square, 

29.  In  establishing-  the  formulas  for  the  differentiation  of 
the  simple  algebraic  functions  of  an  independent  variable,  we 
find  it  convenient  to  begin  with  the  square.  The  object  of 
this  article  is,  therefore,  to  express  dix^)  in  terms  of  x  and 
dx. 

We  first  deduce  a  relation  between  two  values  of  the  de- 
rivative of  the  function  and  the  corresponding  values  of  the 
independent  variable  ;  for  this  purpose,  we  assume  two  values 
of  the  variable  having  a  constant  ratio  m.     Thus,  if 

z^:^mxy  ^'  =  m^  x^. 

Differentiating  by  equation  (2),  Art.  21, 

dz  =  m  dxy  and  diz^)  =  n^  d{x^) ; 

dividing,  we  obtain 

d^z')  d{x') 

— —  =  m  — — . 
az  ax 

Whence,  dividing  hy  z  —  m x\o  eliminate  m^  we  have 


_i     d{^)    _\_     d{x') 
s'   dz      ^  X  '    dx 


(I) 


24  ALGEBRAIC  FUNCTIONS.  [Art.  29. 

The  derivatives  -^-  and  -^    are,  by  Art.  23,  functions 

of  z  and  of  x  respectively,  independent  of  the  values  of  dz 
and  dx\  moreover,  equation  (i)  is  true  for  all  values  of  x 
and  z,  these  quantities  being  entirely  independent  of  each 
other,  since  the  arbitrary  ratio  m  has  been  eliminated.  There- 
fore, either  of  these  quantities  may  be  assumed  to  have  a 
fixed  value,  while  the  other  is  variable  ;  hence  it  follows 
that  the  value  of  each  member  of  this  equation  must  be  a 
fixed  quantity,  independent  of  the  value  of  x  or  of  z.  Denot- 
ing this  fixed  value  by  <:,  we  therefore  write 

i    ^(fl)  _ 
x'    dx    ~^' 

or  dix")  =  cxdx (2) 

30.    To  determine   the  unknown  constant  c,  we    apply  this 
result  to  the  identity 

{x-\-/if  =  x'  +  2hx  +  h\ 

Differentiating  each  member  (Art.  22)  by  equation  (2),  we  have 

c  {x  +  h)  d{x  +  h)  =^  cxdx-h  2h  dx ; 

since  d{x  +  h)  =  dx^  this  equation  reduces  to 

chdx  =^  2k  dxy 
or  {c  —  2)  h  dx  =  o. 

Now,  since  k  and  dx  are  arbitrary  quantities,  this  equation 
gives 

c  =  2; 

this  value  of  c  substituted  in  (2)  gives 

d{x'')  =  2x  dx (a) 

That  is,  t/ie  differential  of  the  square  of  a  variable  equals 
twice  the  product  of  the  variable  and  its  differential. 


§   IV.]  THE     SQUAEE.  2$ 

31.  Employing  the  derivative  notation,  this  result  may 
also  be  expressed  thus  : — 

If  /{^)=A  f\x)  =  2X, 

This  derivative  is  negative  for  negative  values  of  x,  there- 
fore, for  these  values,  x"^  is  a  decreasing  function,  as  already 
mentioned  (Art.  lo)  in  connection  w^ith  the  curve  illustrating 
this  function. 

Since  X  and  dx  are  arbitrary,  we  may  substitute  for  them 
any  variable  and  its  differential.  Equation  {a)  therefore  en- 
ables us  to  differentiate  the  square  of  any  variable  whose 
differential  is  known.     Thus, — 

d{^x  -  if  =  2(5^'  -  3)  sdx  =  io(5^  -  3)  dx. 

Again,  d{a  x^  +  d  xf  =  liax"  ^bx)  d(a  x'  -\-bx) 

=  2{ax''  -f  bx)  {2ax  +  b)  dx. 


The  Square  Root 

32>  To  derive  the  differential  of  the  square  root,  we  put 

y  =  |/^, 
whence  y^  ^=  x] 

differentiating  by  {a),     2y  dy  =  dx, 


.        dx 

dy= 

^       2y 

y=  ^x,.'. 

41/-)  =  ^ 

(*) 


That  is,  tke  differential  of  the  square  root  of  a  variable  is 
equal  to  the  quotient  arising  from  dividing  the  differential  of  the 
variable  by  twice  the  given  square  root. 


26  .         ALGEBRAIC  FUNCTIONS.  [Art.  32. 

Thus.  4^(.»_.=)3^_^-_^^, 

or,  using  derivatives, 


dx  ^{a'-*x') 


Examples  IV. 

'^  I.  Differentiate  {7.x  +  3)*,  and  find  the  numerical  value  of  its  rate, 
when  X  has  the  vahie  8,  and  is  decreasing  at  the  rate  of  2  units  per 
second. 

The  differential  required   is  denoted  by  d\{7.x  -\-  3)^],  and  the  rate  by 

7; :  the  rzven  rate  -j—  =  —  2. 

dt  ^  dt  152  units  per  second. 

2.  Find  the  numerical  value  of  the  rate  of  {x"^  —  2;r)%  when  x  =  y, 
and  is  increasing  at  the  rate  of  ^  of  one  unit  per  second. 
Differentiate  the  given  expression  before  siibstitutijig. 

ii  units  per  second. 

V    3.  Find  the  numerical  value  of  the  rate  of   4/(7''  +  x"^),  whenj/  =  7 

and  ;r  =  —  7,  if  J  is  increasing  at  the  rate  of  12  units  per  second,  and 

X  at  the  rate  of  4  units  per  second. 

4  |/2  units  per  second. 

•     4.  If  f{x)  =  x  —  j^{x''-  «'),   find   f'{x),  and    show  that  f{pc)  is   a 

decreasing  function.  _         £ 

/  (^)  -  I  -  ^(^2_^2y 

•J    5.  Differentiate    the    identity    {\/x  +  \/ay  =  x  +  a  ■\-  2  ^dx,    and 
show  that  the  result  is  an  identity. 

6.  Differentiate  y  (fa  _  3^"^)' 

77/*?  co7tstant  factor     ,,  ^ -r.  should  be  separated  from  the  variable 

•'  ^{a^  ~  2ab)  ^  -^ 

factor  before  differentiation.  i  x  —  a 

j^ia^  —  2a  b)     \/{x^  —  2ax) 


§   IV.]  EXAMPLES.  27 

^'    7.  If  fix)  =  (I  +  -r^)^  .  f\x)  =  --^^-r. 

(I  +  xy 


X 


y/lO.    li   fix)  =  ^^ -,  /'(^)  =  I    +      ,^     .,  ,.^ 

Rationalize  the  denominator  before  differentiating. 

J  x^       y*  dy  . 

^  \i.  Given  — i  +  -7^  =  i.  express  -j—  in  terms  of  x,  and  give  the  values 

r     ^~l  ^     dy-\  dy  b  X 

\J\i.  Given  y  =  4^.r,  express  -7—  in  terms  of  x,  also  in  terms  oiy,  and 
eive  the  values  of -~       ^^a  ^L       .  ^_.4/— =  — .   '      ^ 

13.  A  man  is  walking  on  a  straight  path  at  the  rate  of  5  ft.  per 
second;  how  fast  is  he  approaching  a  point  120  ft.  from  the  path  in 
a  perpendicular,  when  he  is  50  ft.  from  the  foot  of  the  perpendicular.? 

Solution  : — 

Let  ;r  denote  the  variable  distance  of  the  man  from  the  foot  of  the 

perpendicular,  so  that  -rr  may  denote  the  known  velocity  of  the  man, 

and  let  a  denote  the  length  of  the  perpendicular  (120  ft.);  then  the 
distance  of  the  man  from  the  point  is  y'(a'  +  jr'^),  of  which  the  rate  of 
change  is  denoted  by 

d\  i^ia'^ -V  xy^  X         dx 

dt  ~  ^{d'  +^^)  dt  ' 

At  the  instant  considered,  x  =  $0  ft.,  while  a  =  120  ft.,  and  —j-  =  —  5  f t 


28  ALGEBRAIC  FUNCTIONS.  [Ex.  IV. 

per  second.  By  substituting  these  values,  we  obtain  —  i||.  Hence  his 
distance  from  the  point  is  diminishing  (that  is,  he  is  approaching  it)  at 
the  rate  of  i|f  ft.  per  second. 

^  14.  If  the  side  of  an  equilateral  triangle  increase  uniformly  at  the 
rate  of  3  ft.  per  second,  at  what  rate  per  second  is  the  area  increasing, 
when  the  side  is  10  ft.  .^  *  *  15 -^3  sq.ft. 

15.  A  stone  dropped  into  still  water  produces  a  series  of  continu- 
ally enlarging  concentric  circles ;  it  is  required  to  find  the  rate  per 
second  at  which  the  area  of  one  of  them  is  enlarging,  when  its  diame- 
ter is  12  inches,  supposing  the  wave  to  be  then  receding  from  the 
centre  at  the  rate  of  3  inches  per  second.  ^ (§  y  j^     jLyrt^  " 

y  16.  If  a  circular  disk  of  metal  expand  by  heat  so  that  the  area  A  of 
each  of  its  faces  increases  at  the  rate  of  o.oi  sq.  ft.  per  second,  at  what 
rate  per  second  is  its  diameter  increasing.?  1  .  1 


,:r-sv.'''^*nr^; 


^  17.  A  man  standing  on  the  edge  of  a  wharf  is  hauling  in  a  rope 
attached  to  a  boat  at  the  rate  of  4  ft.  per  second.  The  man's  hands 
"being  9  ft.  above  the  point  of  attachment  of  the  rope,  how  fast  is  the 
boat  approaching  the  wharf  when  she  is  at  a  distance  of  12  ft.  from  it.? 

5  ft.  per  second. 

"  18.  A  ladder  25  ft.  long  reclines  against  a  wall ;  a  man  begins  to 
pull  the  lower  extremity,  which  is  7  ft.  distant  from  the  bottom  of 
the  wall,  along  the  ground  at  the  rate  of  2  ft.  per  second  ;  at  what  rate 
per  second  does  the  other  extremity  begin  to  descend  along  the  face 
of  the  wall.?  7  inches. 

19.  One  end  of  a  ball  of  thread  is  fastened  to  the  top  of  a  pole  35  ft. 
high  ;  a  man  holding  the  ball  5  ft.  above  the  ground  moves  uniformly 
from  the  bottom  at  the  rate  of  five  miles  an  hour,  allowing  the  thread 
to  unwind  as  he  advances.  What  is  the  man's  distance  from  the  pole 
when  the  thread  is  unwinding  at  the  rate  of  one  mile  per  hour  } 

1^6  ft. 

20.  A  vessel  sailing  due  south  at  the  uniform  rate  of  8  miles  per  hour 
is  20  miles  north  of  a  vessel  sailing  due  east  at  the  rate  of  10  miles  an 


§   IV.]  EXAMPLES.  29 

hour.     At  what  rate  are  they  separating — (or)  at  the  end  of  i^  hours.? 
(3)  at  the  end  of  2^  hours  ? 

Express  the  distances  in  terms  of  the  time.      {a)  J-^V  miles  per  hour. 

21.  When  are  the  two  ships  mentioned  in  the  preceding  example 
neither  receding  from  nor  approaching  each  other  ? 

Put  the  expression  for  their  rate  of  separatioti  equal  to  zero. 

When  /  =  |o  of  an  hour. 

22.  Derive,  by  the  method  employed  in  Art.  29  to  determine  the 
differential  of  the  square,  the  result  d\  —  \  = — ~,c  being  an  unknown 
constant. 


V. 

The  Product, 


33-  Let  X  and  y  denote  any  two  variables  ;  in  order  to 
derive  the  differential  of  their  product,  we  express  xy  by 
means  of  squares,  since  we  have  already  obtained  a  formula 
for  the  differentiation  of  the  square.     From  the  identity 

(x  +  j)'  =  x""  +  2xy  +/' , 
we  derive 

xy^^ix^-yy-^x'-^^f. 

Differentiating,     d{x)')  =  {x  +  y)  (dx  +  dy)  —  x  dx  —  ydy, 

therefore,  d{x y)  ^=  y  dx -V  x  dy {c) 

Since  x  and  j/  denote  any  variables  whatever,  and  dx  and 
dy  their  differentials,  we  can  substitute  for  x  and  y  any 
variable  expressions,  and  for  dx  and  dy  the  corresponding 
differentials.     Thus, 

^[(l  +x')  4/(«'  -  ^')]  =  4/(^>  -  x')2xdx  -  ^~~^f^ 

2d  —  ;^x^  —  I 

X  dx. 


i/(d  -  x') 


30  ALGEBRAIC  FUNCTIONS.  [Art.  34. 

34.  Formula  {c)  is  readily  extended  to  products  consist- 
ing of  any  number  of  factors.  Thus  X^tx^  x^x^,  .  .  .  xp  denote 
the  product  of  /  variable  factors,  then 

d(x^x^  x^ x^  —  x^x^'"Xp  dx^  +  x^  d{x^  x\  •  •  -xj) 

< 

=  x^x^'  •  -Xpdx^+XiX^'  •  'Xpdx^  +  XyX^d{x^'  •  -Xp) 

=  x^x^' '  'XpdXi+x^x^' '  'Xpdx^'  •  •  +XiX^'  •  'Xp_^dxp.     .  {d) 


The  Reciprocal, 

35.  The    differential    of   the    reciprocal    may    now    be 
obtained  by  means  of  the  implicit  form  of  this  function. 
Denoting  the  function  by  j/,  we  have 


Differentiating  the  latter  equation  by  formula  {c),  we  obtain 

ydx  +  xi 
whence  dy=^  — 


ydx  +  xdy  =  o, 
ydx 
~x"' 


substituting  the  value  of  /, 


4)—^^ •    •('^3 


X 


Formula  {d)  enables  us  to   differentiate  any  fraction  of 
which  the  denominator  alone  is  variable  ;  thus, 


Ja  +  b\  dx 

\a-\-x/  ^        Ha'\-xY 


§  v.]  THE  QUOTIENT.  3^ 


The   Quotient, 

36>  By  the  term  quotient,  as  used  in  this  article,  we  mean 
a  fraction  whose  numerator  and  denominator  are  both 
variable.  In  deriving  its  differential,  the  quotient  is  re- 
garded as  the  product  of  its  numerator  by  the  reciprocal 
of  its  denominator.     Thus,  applying  formulas  (c)  and  {d\ 


M^ 


ii) 


dx  X  dy 
■-y-'J-' 
ydx  —  xdy    ■ 


yi  y 


W 


It  will  be  noticed  that  the  negative  sign  belongs  to  the 
term  which  contains  the  differential  of  the  denominator. 

As  an  illustration  of  the  application  of  this  formula,  we 
have 

I2X  — a\  __  2{x'-^B)  — 2x(2x  —  a)       __     b-vax  —  x^ 

Formula  (e)  is  to  De  used  only  zvhen  both  terms  of  the 
fraction  are  variable  ;  for,  when  the  numerator  is  constant,  the 
fraction  is  equivalent  to  the  product  of  a  constant  and  the 
reciprocal  of  a  variable,  and,  when  the  denominator  is 
constant,  it  is  equivalent  to  the  product  of  a  constant  by  a 
variable  factor.     Thus,  if  it  be  required  to  differentiate  the 

fraction ,  the  use  of  formula  (e)  may  be  avoided  by  first 

making  the  transformation, 

x^^-a^      X     a 

=-  +  -; 

ax         a    X 


ALGEBRAIC  FUNCTIONS.  [Art.  36. 


since,  in   this   form,  one  term  of  each  fraction   is  constant 
Hence, 

dx    adx 


\  ax    J  ~~    a        x^  ' 


The  Power. 

37-  To  obtain  the  differential  of  the  power  when  the 
exponent  is  a  positive  integer,  suppose  each  of  the  variables 
x^x^x^"'Xp  in  formula  {c'\  Art.  34,  to  be  replaced  by  x. 
The  first  member  contains/'  factors,  and  the' second/  terms  ; 
the  equation  therefore  reduces  to 

d{x^)=px^''  dx (i) 

Next,  when  the  exponent  is  a  fraction,  let 

y  =  xif  then  f  =  x^ ; 

differentiating  by  (i),  /  and  ^  being  positive  integers,  we  have 

^y  ^  dy=px^~^dx, 

p  x^~^ 
therefore,  dy=--  ——j  dx, 

q  f 

Substituting  the  value  of  y. 


-. dx  —  -x'' 

^    x^-\  ^ 

Again,  when  the  exponent  is  negative,  we  have 


d(xi)=  —  ' dx=:-x''        dx (2) 

1    x^-\  1 


'    -^ 


§  v.]  THE   POWER.  33 

Differentiating  by  formula  {d),  Art.  35,  we  obtain 

d{x      )=—   -^, 

and,  since  7n  is  positive,  we  have,  by  (i)  or  (2), 

_               inx"^'  ^  dx  _      ,   r 

d{x    '")  =  — -^, ——mx    "•    V/.r.     .     .     .  (3) 

Equations  (i),  (2),  and  (3)  show  that,  for  all  values  of  w, 

d{xr)  —  nx''-^dx (/) 

By  giving  to  71  the  values  2,  J,  and  —  i,  successively, 
it  is  readily  seen  that  this  more  general  formula  includes 
formulas  {a),  (b)  and  {d). 

38.  It  is  frequently  advantageous  to  transform  a  given 
expression  by  the  use  of  fractional  or  negative  exponents, 
and  employ  formula  (/)  instead  of  formulas  {b)  and  {ci). 
Thus, 

^[(-^-2Py^]  =  ^(^'  -  ^^T'  =  S{a--  2x')-'xdx, 
and        d    —-, -3  \=^d{a-\-x^-^    =  —  %(a  +  x) -^dx. 

When  the  derivative  of  a  function  is  required,  it  may  be 
written  at  oiice  instead  of  first  writing  t-he  differential,  since 
the  former  differs  from  the  latter  only  in  the  omission  of 
the  factor  dx^  which  must  necessarily  occur  in  every  term. 
Thus,  given 

--  ^^— r^  =  ;ir(l-h;r')-^ 


we  derive    ^  =  (i  +^')-^  -  \x{\  -^x")-'^ .  2x  =  ^f-^g 


34  ALGEBRAIC  FUNCTIONS.  [Ex.  V. 


J 


Examples  V. 


1.  From  the  identity  xy  =  \{x  ■\-  yY  —  \{x — yY  derive  the  formula  foi 
differentiating  the  product. 


^          .        «  +  bx  +  cx"^ 
ifferentiate . 


>  2.  D 

X 

Put  the  expression  in  the  form  —  +  d  +  e  x.  ie rAdx. 

\ 

3.  Find  the  derivative  of 

J  =  ^^rzrp'     See  remark.  Art.  zi,  _  ==  (^»  ^  ^.)  ___^. 


xj    4.  J  =  Sf{x^  -  «'). 


5-J^  = 


dy  3.r' 

dx  ~     (a^  —  .rO'    * 


n/  6.  J/  =  (I  +  2jr^)(i  +  4jr»).  -£  =  ^{i  +  3x+  io;r»). 


^^r 


^  8.  ^  =  (1  +  .if  (I  +  xy.  ^=  4(1  +  ^y  (I  +  -^')(i  +  ■*•  +  2^'). 


9.  J  =  (I  +  ;r"')"  +  (I  +  x")"*. 

~-  =  mn[{l  +  ;r'»)*-'^-'«-^  +  (l  +  jr^")-" -';»:«-»]. 


/                 ^5  _  2a^ 
V  10.  y  = . 


/  a  —X 


dy 

=  I 

+ 

a^ 

dx 

{x-ar 

±. 

a  +  X 

§  v.]  EXAMPLES.  35 


12.  y 


dx      x'  ^{x""  —  ay 


y  ab  r^       y,         o  dy  ab         ix^  —  a^ 


V 


U.  ^  =  ^-^  +  ^— ^•.  If  =  i[(i  -  -)-^  -  (I  4-  x)-ij. 


^  l6.  J  =  («  +  -r)^  (^  —  ;ir)*.r'. 

dv 

-J-=x{a  +  xy  {b  —  xY  \ia  b  -{■  {^b  —  6d)  x  —  gx""]. 

I  ;r"  4-  I  dy  2nx'^-'^ 

i8.  jj/  =  (3^  +  2a,r)3  (^  ~  ax),  '/'  —  —  S^""-*"  '^(3'^  +  ^ax). 


v/19. 


{2a  X  —  x'^y 


Put  in  the  form  {a"  —  b'''){7.ax  —  ;r')  -*.     ~£,  =  sC^'  —  P^ 
yj  20.  y  = 


V(^-'  -  -r') 


bx 


^^'  ^~  ^{2ax  —  xy 

/22.J=|/^. 
/ 


—  t/ 

'{2ax 

-  ^')«  • 

^J. 

a' 

rt'^ 

{a^ 

_^')r 

dy 

n 

^.r 

dx- 

(2a  X 
I 

-:r')a- 

dx~  ii^x)^(i  —xy 


23.  y  = 


^{a^  +  x^)  —  X 


r>     .       ,^        ,      ,  dy       I  r  a""  +  2X''  "1 

fiattonahze  the  deno?mnator.  -y-  =  --r-    — ,,  „  - — ^  -r  2x  \. 

dx      a^\_  \/{a^  +  ^')  J 


36  ALGEBRAIC  FUNCTIONS.  [Ex.   V. 


v 


24.  Two  locomotives  are  moving  along  two  straight  lines  of  railway 
which  intersect  at  an  angle  of  60°  ;  one  is  approaching  the  intersection 
at  the  rate  of  25  miles  an  hour,  and  the  other  is  receding  from  it  at  the 
rate  of  30  miles  an  hour ;  find  the  rate  per  hour  at  which  they  are 
separating  from  each  other  when  each  is  10  miles  fro;ii  the  intersection. 

2\  miles. 

\ ;  25.  A  street-crossing  is  10  ft.  from  a  street-lamp  situated  directly 
above  the  curbstone,  which  is  60  ft.  from  the  vertical  walls  of  the 
opposite  buildings.  If  a  man  is  walking  across  to  the  opposite  side  of 
the  street  at  the  rate  of  4  miles  an  hour,  at  what  rate  per  hour  does 
his  shadow  move  upon  the  walls — {pi)  when  he  is  5  ft.  from  the  curb- 
stone ?  (/i)  when  he  is  20  ft.  from  the  curbstone  ? 

/  (a)  96  miles  ;  {p)  6  miles. 

26.  Assuming  the  volume  of  a  tree  to  be  proportional  to  the  cube 
of  its  diameter,  and  that  the  latter  increases  uniformly  ;  find  the  ratio 
of  the  rate  of  its  volume  when  the  diameter  is  6  inches  to  the  rate 
when  the  diameter  is  3  ft.  ■^. 

27.  If  an  ingot  of  silver  in  the  form  of  a  parallelopiped  expand 
lo^oo  P^-rt  of  each  of  its  linear  dimensions  for  each  degree  of  tempera- 
ture, at  what  rate  per  degree  of  temperature  is  its  volume  increasing 
when  the  sides  are  respectively  2,  3,  and  6  inches  ? 

If  X  denote  a  side,  dx  may  be  assumed  to  denote  the  rate  per  degree  of 
temperature.  -^-^  of  a  cubic  inch. 

28.  Prove  generally  that,  if  the  coefficient  of  expansion  of  each 
linear  dimension  of  a  solid  is  k,  its  coefficient  of  expansion  in  volume 

is  T,k. 

Solution  : — 

Let  X  denote  any  side  ;  then,  if  V  denote  the  volume,  we  shall  have 
V=  cx^',  c  being  a  constant  dependent  on  the  shape  of  the  body. 

Therefore  dV  =  y  x"  dx ; 

or,  since  dx  —  kx, 

dV=  zkcx^  =  3/^  V, 


CHAPTER   III. 

The  Differentiation  of  Transcendental  Functions. 


VI. 

The  Logarithmic  Function. 

39.  In  this  chapter,  the  formulas  for  the  difFerentiation  of 
the  simple  transcendental  functions  are  to  be  established. 

We  begin  by  deducing  the  differential  of  the  logarithmic 
function,  employing  the  method  exemplified  in  Art.  29. 

The  symbol  log;ir  is  used  in  this  article  to  denote  the  loga- 
rithm of  X  to  any  base,  and  log^;ir  is  used  when  we  wish  to 
designate  a  particular  base  b. 

Let         z  =  in  X, 
differentiating  by  Art.  21 

dz  =  vt  dx, 

whence 

Multiplying  hy  z  =  mx,  to  eliminate  m,  we  obtain 

h    d{\ogz)  _      d{\ogx) 
F^        dz       -""       dx ^^^ 

r,>t       1     .       .  d(\o£[,z)        ^diXoQ-x)  ,       .  . 

The  derivatives,        ,        and        ,  — ,  are,  by  Art.  23,  func- 


log  z  =z  log  m  +  log;ir, 

t.  21, 

and 

d{\ogz) 
dz       - 

d{\ogz)  =  d(}ogx)\ 

d(\ogx) 
m  dx 

38  TRANSCENDENTAL  FUNCTIONS,  [Art.  39. 

tions  of  z  and  of  x  respectively,  independent  of  the  values  of 
dz  and  dx\  moreover,  equation  (i)  is  true  for  all  values  of 
X  and  ^,  these  quantities  being  entirel}^  independent  of  each 
other,  since  the  arbitrary  ratio  m  has  been  eliminated.  Hence, 
in  equation  (i),  one  of  the  quantities,  x  or  z,  may  be  assumed 
to  have  a  fixed  value,  while  the  other  is  variable  ;  whence  it 
follows  that  the  members  of  this  equation  have  a  fixed  value 
independent  of  the  values  of  x  and  z ;  we  therefore  write 

^(l02:;ir)  ,  . 

X T^ — -  =  a  constant (2) 

dx  ^ 

This  constant,  although  independent  oi  x,  may  be  dependent 
on  the  value  of  the  base  of  the  system  of  logarithms  under 
consideration.  Denoting  the  base  of  the  system  by  b^  we 
therefore  denote  the  constant  by  B^  and  write  equation  (2) 
thus, — 

41og,^')  =  — T- (3) 


4-0.  To  determine  the  value  of  B,  we  establish  a  relation 
between  two  values  of  the  base  and  the  corresponding  values 
of  this  unknown  quantity. 

Denoting  another  value  of  the  base  by  a,  and  the  corre- 
sponding value  of  the  unknown  constant  by  A^  we  have 

A^Oga^)    =    -^ (4) 

The  relation  sought  may  now  be  obtained  by  differentiat- 
ing, by  means  of  (3)  and  (4),  the  identical  equation 

log^;f  =  log^^  log^;ir,* (5) 

*  This  identity  is  most  readily  obtained  thus, — by  definition 


§  VI.]  THE  LOGARITHM.  39 

Adx       ,         ,Bdx 
thus  obtaining  — —  =  log^  b  — —  , 

X  X 

or  BXoZab^A, 

hence  ^^Za^^  ~  ^» 

that  is,  A   is  the  logarithm  to  the  base  a  of  h^ ;  whence  we 
have 

b^  ^  a^ (6) 

Now,  it  is  obvious  that  the  value  of  a^  cannot  depend 
upon  b,  hence  equation  (6)  shows  that  the  value  of  b^  likewise 
cannot  depend  upon  b\  b^  must,  therefore,  have  a  value 
entirely  independent  of  b.  Denoting  this  constant  value  by  e, 
wc  write 

3^  ==  £ (7) 

Adopting  this  constant  as  a  base,  and  taking  the  loga- 
rithms of  each  member  of  equation  (7),  we  have 

Blo^^b  =  I, 
I 


whence  B 


\og,b' 
Introducing  this  value  of  B  in  equation  (3),  we  obtain 

In  this  equation,   the  differential  of  a  logarithm  to  any 
given  base  is  expressed  by  the  aid  of  the  unknown  constant  e, 

41.  The  constant  e  is  employed  as  the  base  of  a  system  of 

taking  the  logirilhra  to  the  base  a  of  each  member,  we  have 

logax  =  logbx  loga<J. 


40  TRANSCENDEN'TAL  FUNCTIONS.  Art.  4I. 

logarithms,  sometimes  called  natural  or  hyperbolic,  but  more 
commonly  Napierian  logarithms,  from  the  name  of  the  in- 
ventor of  logarithms.  Hence  e  is  known  as  the  Napierian 
base. 

Putting  /^  =  e  in  formula  {g)  we  derive 

'/(log.^)  =  ^ (^') 

The  logarithms  employed  in  analytical  investigations  are 
almost  exclusively  Napierian.  Whenever  it  is  necessary,  for 
the  purpose  of  obtaining  numerical  results,  these  logarithms 
may  be  expressed  in  terms  of  the  common  tabular  logarithms 
by  means  of  the  formula, 

which  is  derived  from  equation  (5),  Art.  40,  by  writing  10  for 
a  and  e  for  b.  The  value  of  the  constant  log,^^  will  be  com- 
puted in  a  subsequent  chapter. 

Hereafter,  whenever  the  symbol  log  is  employed  without 
the  subscript,  logg  is  to  be  understood. 


The  Logarithmic  Ctirve. 
4-2-.  The  curve,  corresponding  to  the  equation 

y  =  loge-^ (l) 

is  called  the  logarithmic  curve. 

Y  y  The  shape  of  this  curve   is  indi- 

cated in  Fig.  4.     It  passes   through 
the   point  A   whose  coordinates  are 
^      ^  ^  (i,  o),  since 


Fig.  4.  log  I  =  o. 

Since  we  have,  from  formula  {g'\ 


§VI.]  LOGARITHMIC  DIFFERENTIATION.  4 1 

dy        \ 

the  value  of  tan^  at  the  point  A  is  unity,  and  therefore  the 
tangent  line  at  this  point  cuts  the  axis  oi  x  at  an  angle  of  45°, 
as  in  the  diagram.     We  have  from  equation  (2), 

when  -^  >  I  tan«/)  <  i, 

and  when  ^  <  i  tan<?!>  >  i  ; 

the  curve,  therefore,  lies  below  this  tangent,  as  shown  in 
Fig.  4. 

The  point  (e,  i)  is  a  point  of  the  curve  ;  let  j5.  Fig.  4,  be 
this  point,  then  OR  will  represent  the  Napierian  base,  and 
B  R  =  I.     Since 

OA  =1,  and  AR>  BR, 

OR>2', 

that  is,  the  Napierian  base  e  is  somewhat  greater  than  2. 

The  quantity  e  is  incommensurable :  the  method  of  com- 
puting its  value  to  any  required  degree  of  accuracy  is  given 
in  a  subsequent  chapter. 


Logarithmic  Differentiation. 

43,  The  differential  of  the  Napierian  logarithm  of  the 
variable  ,r,  that  is  the  expression  -^ ,  is  called  the  loga- 
rithmic differential  of  x. 

When  X  has  a  negative  value,  the  expression  log;r  has  no 
real  value;  in  this  case,  however,  log(— ;r)is  real,  and  we 
have 

/        XT      d{—x)      dx 
^[log  (-  ^)]  =         ^     =  — -. 


42  TRANSCENDENTAL  FUNCTIONS.  [Art.  43. 

This  expression  therefore,  in  the  case  of  a  negative  quantity, 
is  identical  with  the  logarithmic  differential  of  the  positive 
quantity  having  the  same  numerical  value. 

44.  The  process  of  taking  logarithms  and  differentiating 
the  result  is  called  logarithmic  differentiation.  By  means  of 
this  method,  all  the  formulas  for  the  differentiation  of  alge- 
braic functions  may  be  derived^  * 

In  the  following  logarithmic  equations,  it  is  to  be  under- 
stood that  that  sign  is  taken  in  each  case  which  will  render 
the  logarithm  real. 

By  differentiating  the  formulas, — 

log  {±xy)  =  log  {±x)  +  log  (±  j), 

log(±,r«)  =  ;/log(±;t'), 

d(xv)       dx        dy 
we  obtam  =  —  +  -^, 

•  xy  X         y^- 

X      \yJ  X  y 

d{x")  dx 

— '^—  ^^  — ■• 

X"'  X 

These  formulas  are  evidently  equivalent  to  (r),  (r),  and  (/),  of 
which  we  thus  have  an  independent  proof. 

45.  The  method  of  logarithmic  differentiation  may  fre- 
quently be  used  with  advantage  in  finding  the  derivatives  of 
complicated  algebraic  expressions.     For  example,  let  us  take 

71    = r (I) 

Hence,  we  derive 


§  VI.]  THE  EXPONENTIAL   FUNCTION.  43 

log-  «  =  -J  log  {2X)  +  i  log  (l  —  ^')  -  I  log  {x  —  2),    .      .      (2) 

differentiating, 


du    _  _j 3 x_ 2  _  J__ 

11  dx  ~   2x        ^  I  —  y        ^  X  —  2 


(3) 


adding  and  reducing, 

du  —  8;r'  +  24;tr'  —  .r 


therefore 


udx  6  (i  —  y')  (^.r  —  2)jir 

du  —  8,r'  +  24jr"  —  x  —  6 


dx        ^{2xf{i-x'f{x-2f' 

For  certain  values  of  Xy  one  or  more  of  the  quantities  whose 
logarithms  appear  in  equation  (2)  become  negative.  When 
this  is  the  case  these  logarithms  should,  strictly  speaking,  be 
replaced  by  the  logarithms  of  the  numerical  values  of  the 
quantities  in  question ;  this  change  however  would  not  affect 
the  form  of  equation  (3).     See  Art.  43. 

Exponential  Functions, 

4-6.  An  exponential  function  is  an  expression  in  which  an 
exponent  is  a  function  of  the  independent  variable.  The 
quantity  affected  by  the  exponent  may  be  constant  or  vari- 
able.    In  the  first  case,  let  the  function  be  denoted  by 

J  =  ^-^ (i) 

\i  a  is  negative,  a'  cannot  denote  a  continuously  varying 
quantity.  We  therefore  exclude  the  case  in  which  a  has  a 
negative  value,  and  regard  a^  as  a  continuously  varying  pos- 
itive quantity. 

Taking  Napierian  logarithms  of  both  members  of  equation 
(i),  we  have 

log  J  =  X  log^ ; 
differentiating  by  (^0. 


44  TRANSCENDENTAL  FUNCTIONS.  [Art.  46. 

—  =  loGT  a .  dx  ; 

y  ^  ^ 

hence  dy  ^=z  loga.ydjVj 

or  d^a"^)  =  loga.a'^dx {/t) 

Exponential  functions  of  the  form  e-*"  are  of  frequent  occur- 
rence.    Putting  ^  =  €  in  formula  (//),  we  have  * 

d{e^)  =  e^dx; (//) 

hence  the  derivative  of  the  function  e^  is  identical  with  the 
function  itself.  This  function  is  the  inverse  of  the  Napierian 
logarithm ;  it  has  been  proposed  to  denote  it  by  the  symbol 
exp  X. 

4-7.  When  both  the  exponent  and  the  quantity  affected  by 
it  are  variable,  the  method  of  logarithmic  differentiation  may 
be  employed.     Thus,  if  the  given  function  be 

we  shall  have  log-s"  =  x"^  log  {nx)  ; 

differentiating,       —  =  x""  —  +  2x  log  {n  x)  dx, 
hence  d\{n  xy'\  =  {n  x^  x[i  +  2  log  (n x)]  dx. 


1^^,  and 


Examples  VIl 

-^       I.  Given  the  function  j/  —  logj;r;  show  that   — 

dx 

hence  prove  that  the  tangent  to  the  corresponding  curve,  at  the  point 

whose  abscissa  is  e,  passes  through  the  origin. 

Put  a  =.  X  ^=it  in  equation  5,  Art.  40. 


]  VI.]  EXAMPLES.  ■  45 

^    2.  y  =  ;r"  log  X.  -^  =  jr"  ~  ^  (I  +  «  log  x), 

ax 

/  .       .,         .  dy  \ 

3.  j  =  log  (log ^).  -^   — 


dx      X  log  jr 


4.  jF  =  log[log(^  +<^^)].  -       — 


dx       {a  +  <^^")  log(^  +  bx"") 


'i    5.  >/  =  4/.r-log(4/^+  I).  -^ 


dx       2  (  y'jf  4-  i) 

•^  ^  4/.^  —  |/^  ^/.r       (^?  —  x)  j^x 

Put  in  the  form^  log  (  4/^  +  ^/^r)  —  log  (  y'<^  —  |/.r). 

^     7.^  =  logfV(.-.)4-V(--.)].  _^^  =  _-__i___. 

4    8.^  =  log  [.r  +  4/(.r^  ±  ^^)].  -^ 


V    9.  J  =  log—- — - 


dx        ^{x-'  ±  a^) 
dy  _  I 


-/(I  +  x"")  dx        x{i  +  x"") 


^io.y=  iogyO+^)_+4/(i-^)  ^  _ 


4/(1  4-  ■^')  —  |/v  I  —  -^)                                ^-^^  X  4/(1  —  jr'^) 

^11.  y  =  lot;  r^  +  i/Oi-  —  x')\  -4-  =     .,  ..    ----„-. ^    ,  „ 


y  12.  ^= log 


//or        4/(^'^  —  x'')  \_x  +  4/(rt^  —  A'^)]' 


^/ 


l/C^'-*  +  d')—x  dx       X       ^{x''  +  «■■')' 


dy 


i                 1       ,            ,        « (2Ji:  —  ^)                                               ^        ^r*^  +  <?' 
^  14.  y  =  log  (x  —  a) ^ — ^.  -^  = II . 


4^  TRANSCENDENTAL  FUNCTIONS.  [Ex.  VI. 

ax 
y/  ^  A.  dy  I  A- 


Je^  — 


23.  j^  =  log  (e*  +  c-'). 


dx  (I    +  ^)^ 


dx 


17.  jr  =  e'(i  _  ;r»).  ^-  =  ^'(i  -  3-^'  -  ■^'). 


18.    V  —  (x  —  'C^t'^^/LX^.  4L  : 


18.  ^  =  (^  —  3)  e'*  +  4^*'.  -^  =  {2.x  —  5)  e^'  +  \{x  +  i)e*. 


20.  J  =  3«^  -^  =  log«  .  log/5  .  ^"^ .  a'. 

^   21.  y  =^=«'*.  -4-  —  na^'^ .  ^-'»-'  .  \os:a. 

dx 

J                         X                                                                         ^         f^  ( I  —  .r)  —  I 
V     22.  _/  = -^  =  — ^ . 


£*—  I  dx  (t""—  !)'■' 

dy  f^  —  e" 


dx        e^  4-  £" 


V  24.  y  =  ^  '"ff*.  —  =  —  lojy^  fl! .  rt  ""s*. 

^25.^  =  log-_.  ^^FTIT- 

\     26.  J  =;r^.  -J-  =  ^-'  (i  4-  log;r). 

dx 

^■■^       (^-2)*(jc-3)^'  ^^  12(^-2)^  (a:-3)^3^     ' 

6"^^  ^r/.  45. 


§VI.] 


EXAMPLES. 


28.  y 


V  (jc+i)' (x  +  3)' 

^9^y  =  — (^+,)^ 


47 


^  _     i/t3;(ji:''—  Sax -{-120^) 


'(^  +  3)^ 


v/ 


VII. 


77^^    Trigonometric  or  Circular  Functions, 

4-8-  In  deriving  the  differentials  of  the  trigonometric 
functions  of  a  variable  angle,  we  employ  the  circular  measure 
of  the  angle,  and  denote  it  by  0.  Thus,  let  s  denote  the 
length  of  the  arc  subtending  the  angle  in  the  circle  whose 
radius  is  a,  then 


In  Fig.  5,  let  OA  be  a  fixed  line,  and  OP  an  equal  line 
rotating  about  the  origin  O ;  then  P 
will  describe  the  circle  whose  equation 
(the  coordinates  being  rectangular)  is 


x'^-f 


The  velocity  of  the  point  P  is  the  rate  of 

ds 
s,  and  (see  Art.   17)  is  denoted  by  -j* 

which  has  a  positive  value  when  P 
moves  so  as  to  increase  Q.  Let  PP , 
taken  in  the  direction  of  the  motion  of  P,  represent  ds\  then, 
according  to  the  definition  given  in  Art.  25,  PP'  is  a  tangent 
line,  and  PB  and  B  P  will  represent  dx  and  dy^  as  in  Art.  26, 


Fig.  5. 


48  TRANSCEN-DEiVTAL   FUNCTIONS,  [Art.  49. 

49.  We  have  first  to  show  that  the  line  PP\  which  is  a 
tangent  to  the  curve  according  to  the  general  definition  (Art. 
25),  is  perpendicular  to  the  radius. 

Differentiating  the  equation  of  the  circle,  we  have 


xdx^ 

■  ydy=  0 

) 

whence 

tdLXKp- 

_dy  _ 
~  dx" 

X 

y 

« 

Now  (see 

Fig.  5), 

X 

=  tan(9, 

therefore, 

tan</) 

=  —  cot  0  =  tan 

{e±i 

'T). 

or, 

0  = 

-.Q±\7t', 

hence  the  tangent  line  is  perpendicular  to  the  radius. 
Assuming  0  to  be  the  angle  between  the  positive  directions 
of  X  and  ds^  we  have 


The  Sine  and  the  Cosine, 
60-  From  Fig.  5,  it  is  evident  that 

sm  0  =  -,  and  cos  B  =  -', 

a  a 

therefore  ^(sin^)  =  — ,     and     ^(cos0)  =  --.    .     .     »     (i) 

In  equations  (i)  we  have  to  express  dy  and  dx  in  terms  of 
Q  and  dd. 


§  VII.]  THE    TANGENT  AND    THE   COTANGENT.  49 

Again,  from  the  figure,  we  have 

^  =  sin  0.  dsy  and  dx  =  cos  0.  ds  ;* 

substituting  in  equations  (i),  we  obtain 

^(sin  0)  =  sin  0  — ,     and    ^(cos  0)  =  cos  0  — .     ...    (2) 

Since  ^=:e  +  ^n,  and  -  =  ^f 

,  ds 

sin  (p  =  cos  0,     cos  </>  =  —  sin  0,         and         —  =  dO, 

Substituting  these  values  in  equations  (2),  we  obtain 

d{sind)  =  cosddd,       (/) 

and  d{cosO)  =  —  sinddd (y) 

TAe   Tangent  and  the  Cotangent, 

51.  The  differential  of  tan0  is  found  by  applying  formula 
(e)  to  the  equation 

sin0 

tan  Q  = ; 

cos  Q ' 

. ,  r  /.      ^N        cos  0  <^(sin  0)  —  sin  0  <^(cos  &) 

thus,  ^(tan0)  = ^ ^ — ^-^ ^^ ^, 

^         ^  cos  w 

or  ^(tan0)  =  ^^^z=sec'0^a {k) 


*In   Fig.  5,  dx  is  negative  ;  but,  0  being  in   the  second   quadrant,  cos0  is 
lil^ewise  negative. 


50  TRANSCENDENTAL   FUNCTIONS.  [Art.  5 1. 

The  differential  of  cot  0  is  found  by  applying  formula  {k) 
to  the  equation 

COt0  =  tan(i;r  —  Q)\ 
whence        d{c,oX.Q)  —  —  -^^^  =  —  costd'ddd,      .     .     .     (/) 


The  Secant  and  the  Cosecant. 

52.  The  differential  of  sec  0  is  found  by  applying  formula 
(^)  to  the  equation 

sec^  = 


cos^ 


,  ,,         ,       sine  do 

whence  d^sec  d)  =  -^^-^  =  sec  d  tan  Odd.      ,      .     (m) 

The  differential  of  coseca  is  found  by  applying  formula 
(m)  to  the  equation 

cosec  d  =  sec  (i  ^  —  6) ; 

,  7/  ^N  cos  Odd 

whence    d {cosec  d)  = ^-^— -  =  —  cosec  0  cot  0^(?.      .      (n) 

sin  u  ^  ■^ 


The   Versed-Sine. 
S3.  The  versed-sine  is  defined  by  the  equation 
vers  (9=1—  cos  Q  \ 
therefore  ^(vers  (9)  =  sin  a  ^0 f^) 


§  VII.]  EXAMPLES,  51 

Examples  VII. 

1.  The  value  of  ^^(sin^)   being  given,  derive  that  of  d?(cosO)   from 
the  formula 

cosO  =  sin  (^TT  — 0)  ; 
also  from  the  identity 

cos^G  =  I  —  sin^Q. 

2.  From   the  identity  sec'^=  i  +  tan'0,  derive  the   differential  of 
secO. 

3.  From  the  identity  sin2  0  =  2  sin  0  cos  0,  derive  another  by  taking 
derivatives.  cos  2  0  =  cos'  0  —  sin' G. 

4.  From    the  identity  sin  (0  ±  \t:)  =  i  |/2  (sinG  ±  cosG),  derive  an- 
other by  taking  derivatives.  cos (G  ±  i  tt)  =  i  |/2  (cos G  ^  sin  G). 

5.  Prove  the  formulas  : — 

^/(log  sin  G)  =  —  ^(log  cosec  0)  =  cot  G  d^  ; 
//(log  cos  G)  =  —  ^(log  sec  G)  =  —  tan  G  d^  ; 
<f(log  tan  G)  =  —  ^(log  cot  G)  =  (tan  G  +  cot  G)  ^9. 

6.  Obtain  an  identity  by  taking  derivatives  of  both  members  of  the 
equation 

I  —  cos  9 


tani0  = 


sinG 


,  o  ,  «  I  —cos 
\  sec"  i  G  =  — ^^-^ 
*  ^  sin'  e 


7.  ^  =  G  +  sin  G  cos  G.  -jz  =  2  cos"  1 

8.  y  =  sin  G  —  ^sin'G.  -~  =  cos''G. 

_      sinG  /^_i+cos'G 

^*  -^  ~  V(cos  G)-  5"G  -  7(^6)3 ' 


52 


TRANSCENDENTAL   FUNCTIONS.  [Ex.  VI I. 


j  lo.  y  =  \  tan^G  —  tan  0  +  0. 
•*    II.  _y  =  ^tan^e  +  tanQ. 
^  12.  y  =  sine*. 
J  13.  y  —  X  s\n  x^, 
^    14.  J  =  ^•"*. 

>/    1 5.  J  =  tan^*  0  +  log  (cos^  9). 
^  16.  _y  =  log  (tan  0  +  sec  0). 
17.  J  =  logtan(i7r  +  i0). 
^    18.  _>/  =  ;(:+  log  cos  (iTT—x). 


■; 


19.  7  =  log  yCsin  x)  +  log  ^/(cos  ;ir). 


20.  J  =  sin  n  0  (sin  0)*. 


J                     sin;r 
^  21.  J  = 


I  +  tan  X 


\ 

'  22.  J  =  t^'^cosbx, 

4  _  Mcosx  —  b^\nx 

3-  J  —   og  |/  ^cos^  +  bsinx' 


dy 


^0 


=  sec*  0. 


=  e*  cos  e*. 


■J-  =  sin  ;r''  +  ^x"^  cos  ;r'. 


-^  =  log  a  .  «""*  COS  JIT. 


|  =  2tan30. 


^0 


=  sec  0. 


dy  _     I 
^""cos0* 

<^_         2 
~dx~  \  ■\-  tan  y 


^;r 


=  cot  2X. 


dv 

jQ=n  (sin  0)"  - '  sin  (?z  +  I) 0. 

dy       cos^x  —  sin'^r 
dx~  (sin  X  +  cos;*:)''* 

dy 

-^  =  ef''  {a  cos  b X  —  bsmbx). 


-ab 


dy^ 

dx      d^  cos^  X  —  b^  sin' X* 


§  VII.]  EXAMPLES.  53 

^    2^.  y  ~  ^' (q.o's.x —s\T\x).  -^=  — 2fc^sinx 

25.  The  crank  of  a  small  steam-engine  is  i  foot  in  length,  and 
revolves  uniformly  at  the  rate  of  two  turns  per  second,  the  connect- 
ing rod  being  5  ft.  in  length  ;  find  the  velocity  per  second  of  the 
piston  when  the  crank  makes  an  angle  of  45°  with  the  line  of  motion 
of  the  piston-rod ;  also  when  the  angle  is  135°,  and  when  it  is  90°. 

Solution : — 

Let  a,  b,  and  x  denote  respectively  the  crank,  the  connecting-rod, 
and  the  variable  side  of  the  triangle  ;  and  let  0  denote  the  angle  be- 
tween a  ?Lnd  X.  ^.  . 

We  easily  deduce  *  ^*^tf    £^ 

;ir  =  «cos0  +  i/(^'  — rt'sin'»0);  >tc^^' 

(^--^•^^    etc 


whence 


dx  I  ^'sinOcosQ     \d^ 


dO 
In  this  case,  -tj  =  ^n,  a  =  i,  and  ^  =  5. 

When  6  =  45°,  ;^7  = -f-  ft. 

26.  An  elliptical  cam  revolves  at  the  rate  of  two  turns  per  second 
about  a  horizontal  axis  passing  through  one  of  the  foci,  and  gives  a 
reciprocating  motion  to  a  bar  moving  in  vertical  guides  in  a  line  with 
the  centre  of  rotation  :  denoting  by  6  the  angle  between  the  vertical 
and  the  major  axis,  find  the  velocity  per  second  with  which  the  bar  is 
moving  when  0  =  60°,  the  eccentricity  of  the  ellipse  being  \,  and  the 
semi-major  axis  9  inches.     Also  find  the  velocity  when  6  =  90°. 

The  relation  between  0  and  the  radius  vector  is  expressed  by  the  equation 


a(i  —  e^)  r^'-^'^ 


1  u 


^cos9 


9 


When  0  =  60°,  -r-  =  —  12  4/3  TT  inches. 

27.  Find  an  expression  in  terms  of  its  azimuth  for  the  rate  at  which 
the  altitude  of  a  star  is  increasing. 

Solution : — 

Let  //  denote  the  altitude  and  A  the  azimuth  of  the  star,/  its  polar 
distance,  /  the  hour  angle,  and  L  the  latitude  of  the  observer  ;  the 
formulas  of  spherica!  trigonometry  give 


54  TRANSCENDENTAL  FUNCTIONS.  [EX.   VII. 

sin  ^  =  sin  Z  COS /^  +  cos Z  sin/ cos /,     .     .     .     .     (i) 

and  sin/  sin  /  =  sin  ^  cos  h.      ......     (2) 

Differentiating  (i),/  and  Z  being  constant, 

cos  "--f.^  ~  COS  Z  sm/  sm  /, 
(It 

whence,  substituting  the  value  of  sin/  sin  /,  from  equation  (2), 

-—  =  —  cos  Z  sin  A. 
at 

It  follows  that  —  is  greatest  when  sin^  is  numerically  greatest ;  that 

is,  when  the  star  is  on  the  prime  vertical.     In  the  case  of  a  star  that 
never  reaches  the  prime  vertical,  the  rate  is  greatest  when  A  is  greatest. 


VIII. 

The  Inverse   Circular  Functions. 

S^.  It  is  shown  in  Trigonometry  that,  if 

X  =  sin  Qy 
the  expressions 

2n7t-\-0  and  (2;^+ i)  tt  —  <9,     .     .    (i) 

in  which  n  denotes  zero  or  any  integer,  include  all  the  arcs 
of  which  the  sine  is  x\  hence  each  of  these  arcs  is  a  value 
of  the  inverse  function 


Among  these  values,  there  is  always  one,  and   only  one^ 
which   falls  between  —  \n  and  +i7r;  since,  while  the  arc 


§  VIII.]  INVERSE  CIRCULAR  FUNCTIONS.  55 

passes  from  the  former  of  these  values  to  the  latter,  the  sine 
passes  from  —  i  to  + 1  ;  that  is,  it  passes  once  through 
all  its  possible  values. 

Let  0,  in  the  expressions  (i),  denote  this  value,  which  we 
shall  call  the  primary  value  of  the  function. 

65-   In  a  similar  manner,  if 

X  =  cos  9j 

each  of  the  arcs  included  in  the  expression 

2n7r  ±e (2) 

is  a  value  of  the  inverse  function 

cos  " '  ,v. 

One  of  these  values,  and  only  one,  falls  between  oand  n  ; 
since,  while  the  arc  passes  from  the  former  of  these  values 
to  the  latter,  its  cosine  passes  from  +  i  to  —  i  ;  that  is,  once 
through  all  its  possible  values.  In  expression  (2),  let  d  denote 
this  value,  which  we  shall  call  the  primary  value  of  this 
function. 

56,   In  the  case  of  the  function 

cosec  ~  ^  x^ 

the  definition  of  the  primary  value  that  was  adopted  in  the 
case  of  sin~*;ir,  and  the  same  general  expressions  (i)  for  the 
values  of  the  function,  are  applicable. 
In  the  case  of  the  function 

sec~*;tr, 

the  definition  of  the  primary  value  adopted  in  the  case  of 


56  TRANSCENDENTAL  FUNCTIONS.  [Art.  56. 

cos-'-JT  and   expression   (2)   for   the    general   value   of    the 
function  are  applicable. 

Finally,  in  the  case  of  each  of  the  functions 

tan~';ir  and  cot~';i; 

t\iQ  primary  value  {d)  is  taken  between  —^7t  and  -\-^7t,  and 
the  general  expression  for  the  value  of  the  function  is 

njt  +  e (3) 

The  Inverse  Sine  and  the  Inverse  Cosine, 
67-  To  find  the  differential  of  the  inverse  sine,  let 
Q  =  sin-^;tr; 
then  x=sm8,  and  dx  =  cosOdd^ 

dx 
or  dQ=  — ^. 

cos^ 


(I) 


If  B  denotes  the  primary  value  of  this  function ;  that  is,  the 
value  between  —  ^7rand+|-^,  cos0  is  positive.  Hence  the 
upper  sign  in  this  ambiguous  result  belongs  to  the  differential 
of  the  primary  value  of  the  function ;  it  is  therefore  usual  to 
write 

</(sin--^)  =  -^-^^^ {p) 

Since  we  have,  from  expressions  (i).  Art.  54, 

d(2n  TT  +  0)  =  dQ,        and        d\j^2n  +  i)  tt  —  0]  =  —  dQ^ 


Now, 

COS0  =  ±  4/(1  —  sin'e)  =  ±  4/(1  . 

-n 

Vifnpf* 

di-\n-^x\  —            "^^ 

^^.m     X)-  j^^(,  _^.y 

§  VIII.]     THE  INVERSE  SINE  AND  INVERSE  TANGENT.         57 

it  is  evident  that  the  positive  sign  in  equation  (i)  belongs  not 
only  to  the  dillerential  of  the  primary  value  of  sin~';r,  but 
likewise  to  the  differentials  of  all  the  values  included  in 
2mT  +  d\  and  that  the  negative  sign  belongs  to  the  differen- 
tials of  the  values  of  sin~'^  included  in  (2;/+  i)7t  —  0. 

58.  Similarly,  if 

0  =  COS~^Jtr,  ;r  =  cos0; 

_^  dx 

whence  '^^=-:zji^e' 

or  d{coB-'x)  =  —^^^^ (I) 

If  0  denote  the  primary  value  of  the  function  which  in 
this  case  is  between  o  and  tt,  sin  0  is  positive  ;  hence  the  up- 
per sign  in  this  ambiguous  result  belongs  to  the  differential  of 
the  primary  value.     It  is  therefore  usual  to  write 

_  dx 

Since,  from  expression  (2),  Art.  55,  we  have 

d{2n  7r±0)  =  ±d0; 

it  is  evident  that  the  upper  and  lower  signs  in  equation  (i) 
correspond  to  the  upper  and  lower  signs,  respectively,  in  the 
general  expression  2^1  tt  ±0. 

The  Inverse   Tangent  and  the  Inverse  Cotangent, 

69.  Let 

0  —  tan  ~  *  X,  then  x  =  tan  d ; 

differentiating,  we  derive, 

dx 


sec'O 


5^  TRANSCENDENTAL  FUNCTIONS.  [Art.  ^Qk 

But  sec''  0=1+  tan^  0  =  i  +  ;r',  therefore, 

dx 

No    ambiguity  arises  in  the  value  of  the   differential  of 
this  function;  since,  from  expression  (3),  Art. ^56,  we  have 

d{n  TT  +  d)  =  do. 
Similarly,  putting 

e  =  cot  ~^  Xf 

we  derive  ^(cot-'x)  =  ~j^~ (s) 


The  Inverse  Secant  and  Inverse   Cosecant, 
60.  Let 

0=sec~*;i:,  then  ;tr  =  sec0; 

differentiating,  we  derive 

dx 


do  = 


sec  d  tan  0 ' 


But     sec0  =  X,     and     tan0  =  ±  ^/(sec'^  —  i)  =  ±  -y'{x'  —  i), 
therefore, 

jr        -1    \                  dx 
d{sec      x)  = -^-^ 

If  X  is  positive,  and  if  0  denotes  the  primary  value  of  the 
function,  tan0  is  positive.     Hence  it  is  usual  to  write 

dx 


§  VIII.]  THE  INVERSE  SECANT.  59 

When  X  is  negative,  if  0  denotes  the  primary  value  of  the 
function,  which  in  th :s  case  is  in  the  second  quadrant,  tan Q  is 
negative ;  consequently  the  radical  must  be  taken  with  the 
negative  sign.  Hence,  since  x  is  also  negative,  the  value  of 
the  differential  is  positive,  when  the  arc  is  taken  in  the 
second  quadrant. 

In  like  manner  we  derive 

4cosec-^)  =  -^^^^f_^^ («) 

Similar  remarks  apply  also  to  this  differential  when  x  is 
negative. 

The  Inverse    Versed-Sine, 
6r.  Let 

0  =  vers~*-r,         then         x  =  vers0  =  I  —  cos^, 

dx 
and       \  —  X  =^  cos  Q,  .  • .  ^0  =  -: — --, 

*  sm0 

But  sin0  =  |/(i  —  cos'0)  =  4/(2^—  ^),  therefore, 


Illustrative  Examples. 

62-  It  is  sometimes  advantageous  to  transform  a  given 
function  before  differentiating,  by  means  of  one  of  the 
following  formulas : — 

sm~  -;5  =  cosec      -,     cos~  -;5  =  sec      -,    tan      75  =  cot     -. 


60  TRANSCENDENTAL  FUNCTIONS.  [Art.  62. 


Thus,  let  y  —  tan  ~ '  — 


e-^  sin  ;ir 


then  7=  cot"' (e^^sec^tr  +  tan;!;). 

By  formula  {s\ 

dy  _         e--*sec;rtan;ir  —  e--»^secar-+ ^c';r 
dx  ~       sec'';ir  +  2e--^sec;r  tan  ;ir4-«~  ^*  sec^;r* 

multiplying  both  terms  by  e^cos^^r, 

dy       £-^(cos  X  —  sin  x  —  e^) 
dx  ~       I  +  2  e-^  sin  ;ir  +  e^'^      ' 

63.    Trigonometric    substitutions     may     sometimes     be 
employed  with  advantage.     Thus,  let 

y  =  tan 


4/(l+;ir^)+  I 

If  in  this  example  we  put  x  =  tan^,  we  have 

_,     tan^  ,     sin^ 

y  =  tan =  tan  ~    - 


sec  ^  +  I  I  ■\-  cos  0 

=  tan-*(tanJ0)  =  i-^  =  ^tan-';r. 


Examples  VIII. 
V    I.  Derive  from  (/),  (r),  and  (/)  the  formulas: — 


^ftan-^)=^ 


§   VIIL]  EXAMPLES.  6 1 

(;i:\  adx 

a)       X  ^{x^  —  a^) 

2.  Derive  ^(sec  ~^x)  from  the  equation  sec  -^x  =  cos -^  — 

3.  Derive  ^f cot-'  — J  from  the  equation  cot"'— =  tan-*— • 

'  dy  >4jr 

4.  J  =  sin  - '  {2x^),  — 

5.  ^  =  sin-*  (cos;r), 

6.  y  =  sin  (cos" *;r), 

77" 

7.  ^  =  sin-*  (tan;r). 


8.  /  =  cos-*  (2Cos;r). 


9.  j=;rsin-*;r+  4/(1  — ^r"). 

10.  _y  =  tan-*f*. 

11.  y={x^  +  i)  tan-*;r  —  ;r. 

^  V  12.  ^=«''sin-*|^  +  ;r  v'C^'^  -  ^r'^).  ^  =  2  |/(^^  -  jr") 

^     V   13.  ^  =  tan-*j^^;-^. 
^+  I 


dx-  . 

^(1-4^*)- 

dy 
dx^"  ^' 

dy 
dx~ 

X 

v{i-^r 

dy 

sec'jr 

dx       4/(1 

[  —  tan'^^r)* 

^^ 

2sin-r 

^-r 

--^./(i. 

—  4COS='^)* 

^J 
//;»: 

=  sin-*  jr. 

dy 
dx 

I 

-£-+£-"' 

dy 
dx- 

2;rtan-*;ir. 

[4.  y  =  sin 


•  1  — 


dy 

m{\ 

+  -r^) 

dx~ 

-  i^{m^- 

-2);r» 

+  ^* 

dy 

I 

4/2   '  dx       ^{i—2x—x*y 


62  TRANSCENDENTAL  FUNCTIONS.  [Ex.   VIII. 


*> 

V  — 

Lau 

4/(1-^^)- 

i6. 

^  = 

sec- 

V 

i/i7. 

y^ 

sin- 

-1 

v/. 

18. 

y  = 

sin- 

■»  4/(sin  r). 

v/. 

19. 

y  = 

4/(1 

—  x')sin->jr— jr. 

,    »2  4-  ^ 

20.  _y  =  tan-^- 


mx 
I  -x» 


-^  ,     /I  —  cosjtr 

22.  y  =  tan  -  ^  4/ — ; . 

■^  r     I  +  cos;r 

J 


J 


1/  jjrsin-';r       ,  ,  ^ 


24.  y—{x-\-d)  tan-'  |/ 4/(«;f). 


dx~  4/(1  —  ^^)' 
^  I 


dx       ^{cr-x'^y 
dy  a 


dx~  a"  ■Vx''' 


-^  =  1^(1  +cosec^). 


dr 

dx 


;rsir 

l-i^ir 

^(i 

-^^) 

^ 

^/;r       I 

I 

^__ 

2 

^J 


Tx-^' 

^/ 

sin  -';r 

^x- 

~(.-x')«- 

^7 

dx- 

'^""'/l- 

IX. 

Differentials  of  Functions  of  Two  Variables. 

64.  The  formulas  already  deduced  enable  us  to  differen- 
tiate any  function  of  two  variables,  expressed  by  elementary 
functional  symbols ;  the  application  of  these  formulas  is,  how- 


§  IX.]  FUNCTIONS  OF   TWO    VARIABLES.  63 

ever,  sometimes  facilitated  by  a  general  principle  which  will 
now  be  shown  to  be  applicable  to  such  functions. 

The  formulas  mentioned  above  involve  differential  factors 
of  the  first  degree  only.  It  follows,  therefore,  that«the  differ- 
entials resulting  from  their  application  consist  of  terms  each 
of  which  contains  the  first  power  of  the  differential  of  one  of 
the  variables.     In  other  words,  if 

du  =^<l>{x,y)dx-V\\){x,y)dy (l) 

Now,  if  y  were  constant,  we  should  have  ^  =  o,  and  the 
value  oi  dii  would  reduce  to  that  of  the  first  term  in  the  right- 
hand  member  of  (i);  hence  this  term  may  be  found  by  differ- 
entiating u  on  the  supposition  that  y  is  constant^  and  in  like 
manner  the  second  term  can  be  found  by  differentiating  u  on 
the  supposition  that  x  is  constant.  The  sum  of  the  results 
thus  obtained  is  therefore  the  required  value  of  du, 

65.  As  an  example,  let 


u^. 


Were   v  constant,  we   should    have  for   the   value  of  dz,  by 
formula  (/),  Art.  37, 

vu'"—'^du\ 
and,  were  u  constant,  we  should  have,  by  formula  (Z^),  Art.  46, 

log  u  .  u"^  dv] 
whence,  adding  these  results, 

ds  ^  «"  -  \v  du-\-u  log  ti  dv\ 


64  DIFFERENTIATION.  [Art.   65. 

Although  this  result  has  been  obtained  on  the  supposition 
that  u  and  v  are  independent  variables,  it  is  evident  that  any 
two  functions  of  a  single  variable  may  be  substituted  for  u 
and  V.     Tiius,  if 

u  =  nx  and  z/  =  4r', 

we  have  z  =  {n  x^y  * 

and,  on  substituting, 

ds  =  {n xy -'^{x'^ndx+  n x  log {n x)  .  2x dx), 
=  x(nxy[i-\-2\og{nx)']dx, 

which  is  identical  with  the  expression  obtained  in  Art.  47,  for 
the  differential  of  this  function. 

Examples  IX. 

I.  u  =  xy  e*  +  «y.  du=e'+^yly(i  +  x)^;r  +  x(l  +  2y)dy]. 

2.u  =  log  tan^.  du=2^-^^^^Z^. 

i/^sin2  — 

y 

3.  «  =  logtan-':^.  ^^_  ^^^-^- -^^J 


y  C^'^+jOtan-^^ 


^^^*^-±^^y 


^^^-b--y--2  4/(.r7)1  ^ydx  ^\x-y-^2  ^/{xy)\  i/xdy 
^V{.xy){x^yy 

J    J    r    ^^        ""y ■^■^_     y'^^        .    x{xdy-ydx)^ 


§  IX.]  MISCELLANEOUS  EXAMPLES  65 


y  / 


dy  _ 
dx 

X  + 

2 

2(1  + 

-)' 

dy 

(^' 

-b'')x 

dx 

{a^-x\ 

)'^(^*^-. 

.')* 

Va{Vx- 

■  Va) 

/  / 

'         9.  Given  x  =  r  cos  9,  and  j  =  r  sin  Q;  eliminate  6  and  find  dr ;  also 
eliminate  r  and  find  dB, 

.        X  dx -{■  y  dy         ,    ..       xdy—ydx 
dr  =      ..  ,   ,     .;.  ,  and  dO  =  —  3   ,  -^ ^     . 

Miscellaneous  Examples. 

^'  -^    V  +  Vx'  dx      2  Vx  V(a  +  x)(Va  +  Vx)" ' 

,     4.,=  (Vx-2Va)V{Va+Vx).         |=^^(^,%^,). 
^  _  (o:-  i)(g'  4-  i)e'  dy  _  £' {x  s""'  -  2x  s^  +  2s'  —  x) 

^  6.  ^  =  log-^^ fr  +  i  tan    ^x.  -f-  =  7 — ■ TT — ^ — Tx  • 

■^  ^   (i  +  ^)«  dx       {i+x){i-^x) 

/  2sin~*^     ,   ,      I— ^  4^       2Jtsin"^^ 


66  DIFFERENTIATION.  [Ex.   IX, 


^ 


V    1 1.  JF  =  ^,log ^ -'  —  V  [a   —  X 


dy_        V  (g"  -  x^) 
dx~      ^        X 


\                  (i  —  :v'^)8sm~* 
12.  V  =  -'^ '- 


X 
X 


dy        1—  X        1  -{•  2x^    .,  2\   •    -1 

~  = ' i —  Vii  —  X)  sm   ^x, 

ax  X  X  ^  ^ 


n)     .,    .-w./^-^Q^^  ^^ 


13.  J 


log/f^ 


cos:^?  dx      sinjc* 


4    I4.7  =  tan-r^^.tanf]. 


dx~  2{a  +  b  cos  jc) ' 


J  _t        I  /^  2 

^    15.7=  sec  ^  — ^ .  -f  = — — ^ 

2^—1  d!'.;C  1/  (l   —  ^') 


J     16.  J  =  COS     ^  -^ 


^  _       2nx'' 


X'"  +1  ^-  ;v'"  +  I 

dy_ 3c_ 

'dx  ~  V  [b'  -{a-  xy^  • 

Jo  -1  i/i— -^  dy      \/ (1  —  x) 

^    18.  y  =  cos^jt:  —  2i/ .  ^  —  JL^ J.  ^ 

^   i+x  dx        (^j^x)^ 

£/j^  logarithmic  differentials. 


CHAPTER    IV. 
Successive  Differentiation. 


X. 

Velocity  and  Acceleration, 

66.  If  the  variable  quantity  x  represent  the  distance  of  a 
point,  moving  in  a  straight  Hne,  from  a  fixed  origin  taken  on  the 
line,  the  rate  of  x  will  represent  the  velocity  of  the  point. 

Denoting  this  velocity  by  v^  we  have,  in  accordance  with  the 
definition  given  in  Art.  17, 

dx  ,  . 

"'^w ('> 

In  this  expression  the  arbitrary  interval  of  time  dt  is  re- 
garded as  constant,  while  dx,  and  consequently  Vj^,  is  in  gen- 
eral variable.  Differentiating  equation  (i)  we  have,  since  dt 
is  constant, 

dt 

The  differential  of  dx,  denoted  above  by  d{dx)y  is  called  the 
second  differential  o{  X  ;  it  is  usually  .written  in  the  abbreviated 
form  d^'x,  and  read  "  d-second  xJ'  The  rate  of  Vx  is  therefore 
expressed  thus : — 

dvjc  _d^x 
It      (dlf' 


68  SUCCESSIVE  DIFFERENTIATION.  [Art.  C}6, 

The  rate  of  the  velocity  of  a  point  is  called  its  acceleration^ 
and  is  usually  denoted  by  a  ;  hence  we  write 


the  marks  of  parenthesis  being  usually  omitted  4n  the  denomi- 
nator of  this  expression. 

67.  When  the  space  x  described  by  a  moving  point  is  a 
given  function  of  the  time  /,  the  derivative  of  this  function  is, 
by  equation  (i),  an  expression  for  the  velocity  in  terms  of  /. 
The  derivative  of  the  latter  expression,  which  is  called  the 
second  derivative  of  x,  is  therefore,  by  equation  (2),  an  expres- 
sion for  the  acceleration  in  terms  of  /. 

A  positive  value  of  the  acceleration  a  mdicates  an  algebraic 
increase  of  the  velocity  v,  whether  the  latter  be  positive  or 
negative ;  and,  on  the  other  hand,  a  negative  value  of  a  indi- 
cates an  algebraic  decrease  of  the  velocity. 

68.  As  an  illustration,  let  x  denote  the  space  which  a  body 
falling  freely  describes  in  the  time  /.  A  well-known  mechanical 
formula  gives 

^  =  W' 0) 

dx 

Hence  we  derive  Vx—-r=gti (2) 

ai 

J  dvx      d'^x  ,  X 

^"'^  '^'=-^=^=^- ^3) 

In  this  case,  therefore,  the  acceleration  is  constant  and  posi- 
tive, and  accordingly  v^^  which  is  likewise  positive,  is  numeri- 
cally increasing. 

69.  When  the  velocity  is  given  in  terms  of  x,  the  acceleration 
can  readily  be  expressed  in  terms  of  the  same  variable,  as  in 
the  following  example. 


§  X.]  VELOCITY  AND  ACCELERATION.  69 

Given  Vx—2^mx\ 

dvx  dx 

whence  -7—  =  2  cos  x  -j- ; 

at  at 

that  is,  ^'^  =  2  cos  x,Vx  =  ^  cos  jr  sin  ;ir  =  2  sin  2x, 

The   general  expression  for  a^,  when  v^  is  given  in  terms 
of  x^  is 

_  dvx  _  ^-2^^  ^^  _       dvx  _  I   ^(t^x)  /  \ 

^~~  dt        dx  dt  ~    ^  dx    '  2    dx 


Component   Velocities  and  Accelerations, 

70.  When  the  motion  of  a  point  is  not  rectilinear  but  is 
nevertheless  confined  to  a  plane,  its  position  is  referred  to  co- 
ordinate axes  ;  the  coordinates,  x  and  y,  are  evidently  functions 

of  /,  and  the  derivatives  —-  and  -— ,    which  denote  the  rates 

dt  dt 

of  these  variables,  are  called  the  cojnponent  or  resolved  velocities 
in  the  directions  of  the  axes.  Denoting  these  component  veloci- 
ties by  Vx^nd  Vyy  we  have 

dx         .  dy 

t..  =  ^,  and   v,=f^. 

Again,  denoting  by  s  the  actual  space  described,  as  measured 

from  some  fixed  point  of  the  path,  s  will  likewise  be  a  function 

ds 
of  /,  and  the  derivative  -y    will  denote  the  actual  velocity  of 

at 

the  point.  (Compare  Art.  48.)  Now,  the  axes  being  rectangu- 
lar, and  0  denoting  the  inclination  of  the  direction  of  the  mo- 
tion to  the  axis  of  x,  we  have 

dx  =  ds  cos  ^,  and   dy  =  ds  sin  0. 
-,  dx       ds  .         A   dy       ds    ,    J. 


70  SUCCESSIVE  DIFFERENTIATION,  [Art.   /Q 

or  Vx  —  V  COS  ^,  and   Vy  =v  sin  ^. 

Squaring  and  adding, 

The  last  equation  enables  us  to  determine  from  the  component 
velocities  the  actual  velocity  in  the  curve. 

71.  If  we  represent  the  accelerations  of  the  resolved  mo- 
tions in  the  directions  of  the  axes  by  ajc  and  ofy,  we  shall  have, 
by  Art.  66, 

a.=  ^    and   .,  =  ^  . 

These  accelerations,  a^  and  a^^  will  be  positive  when  the  re- 
solved motions  are  accelerated  in  the  positive  directions  of  the 
corresponding  axes ;  that  is,  when  they  increase  a  positive  re- 
solved velocity,  or  numerically  decrease  a  negative  resolved 
velocity. 


Examples  X. 

J    I.  The  space  in  feet  described  in  the  time  /  by  a  point  moving  in 
a  straight  line  is  expressed  by  the  formula 

^  =  48/  —  16/"; 

find  the  acceleration,  and  the  velocity  at  the  end  of  2  J  seconds ;  also 
iind  the  value  of  t  for  which  z;  =  o. 

Of  =  —  32  ;    z;  =  o,  when  /  =  \\. 

J  2.  If  the  space  described  in  /  seconds  be  expressed  by  the  formula 
jf  =  10  log 


4  +  i' 

find  the  velocity  and  acceleration  at  the  end  of  i  second,  and  at  the 
end  of  1 6  seconds.  When  t=  i,  v=  —  2  and  «  =  f . 


§    X.]  EXAMPLES.  71 

,     3.  If  a  point  moves  in  a  fixed  path  so  that 

s=  Vt, 

show  that  the  acceleration  is  negative  and  proportional  to  the  cube  of 
thp  velocity.  Find  the  value  of  the  acceleration  at  the  end  of  one 
second,  and  at  the  end  of  nine  seconds.  —  :^,  and  —  jj-^. 

"^    4.  If  a  point  move  in  a  straight  line  so  that 

x  =  a  cos  ^Ttty 
show  that  a=  —  ^Tt^x. 

V  5.  If  x  =  a£'  ■{■  d€-\ 
prove  that                          a  =  x. 

J    6.  If  a  point  referred  to  rectangular  coordinate  axes  move  so  that 

X  =  a  cos  /  +  <^        and       y  =  a  sin  f  ■}■  c, 

show  that  its  velocity  will  be  uniform.  Find  the  equation  of  the  path 
described. 

Eliminate  t  from  the  given  equations. 

V  7.  A  projectile  moves  in  the  parabola  whose  equation  is 

y  =  x\.zxia —f — —  x^, 

2  V   cos  Of 

(the  axis  of  _y  being  vertical)  with  a  uniform  horizontal  velocity 

Vx^=  V  cos  a  ; 
find  the  velocity  in  the  curve,  and  the  vertical  acceleration. 

v=  V{V'  —  2gy),  and  a,  =  —g, 
8.  A  point  moves  in  the  curve,  whose  equation  is 
x^  +  ^f  =  a^^ 


7'2  SUCCESSIVE  DIFFERENTIATION-.  [Ex.    X, 

SO  that  Vx  is  constant  and  equal  to  k ;  find  the  acceleration  in  the  di- 
rection of  the  axis  oiy.  ^1^2 

y   9.  If  a  point  move  so  that  v  —  V{2gx);  determine  the  acceleration. 
C/se  equation  (i),  Art.  6g.  ^  =  g- 

\J    10.  If  a  point  move  so  that  we  have 


v"^  =  c  —  M  log  x, 
n. 
yj     II.  If  a  point  move  so  that  we  have 


determine  the  acceleration.  a  =  —  — 


2X 


2^ 


determine  the  acceleration.  a  — -. 

{x'  +  b')^ 

12.  The  velocity  of  a  point  is  inversely  proportional  to  the  square 
of  its  distance  from  a  fixed  point  of  the  straight  line  in  which  it  moves, 
the  velocity  being  2  feet  per  second  when  the  distance  is  six  inches  ; 
determine  the  acceleration  at  a  given  distance  s  from  the  fixed  point. 

-  ~,  feet. 

2S 

y  .... 

13.  The  velocity  of  a  point  moving  in  a  straight  Ime  is  m  times  its 
distance  from  a  fixed  point  at  the  perpendicular  distance  a  from  the 
straight  line  ;  determine  the  acceleration  at  the  distance  x  from  the 
foot  of  the  perpendicular.  ol  =  m^x. 

V      14.  The  relation  between  x  and  /  being  expressed  by 
f  1/ -^=  \/{ax  —  x) —^ayeis  "^ — ; 
find  the  acceleration  in  terms  of  x.  oc= 5  • 

X 

\     15.  A  point  moves  in  the  hyperbola 

/  =P'^X'  +  q' 

in  such  a  manner  that  Vx  has  the  constant  value  c  ;  prove  that 


§  X.]  EXAMPLES.  73 

and  thence  derive  ay  by  equation  (i),  Art.  69. 


7 


<y   =  — — i— 


16.  A  point  describes  the  conic  section 


v^  having  the  constant  value  c  ;  determine  the  value  of  a^. 
Express  Vy  in  terms  of  y^  and  proceed  as  in  Example  15. 


-f 


2  2 
m  c 


-.=  -y 


XI. 

Successive  Derivatives, 


72.  The  derivative  of  f(x)  is  another  function  of  x,  which 
we  have  denoted  by  f\x)  ;  if  we  take  the  derivative  of  the 
latter,  we  obtain  still  another  function  of  .r,  which  is  called  the 
second  derivative  of  the  original  function  f(x\  and  is  denoted 
by/"(^).     Thus  if 

f{x)  =  x\        f'(x)  =  ix\        and         f\x)  =  6x, 

Similarly  the  derivative  of  f'(x)  is  denoted  by  f"'{x),  and 
is  called  the  third  derivative  of  f{x) ;  etc.  When  one  of  these 
successive  derivatives  has  a  constant  value,  the  next  and  all 
succeeding  derivatives  evidently  vanish.  Thus,  in  the  above 
example,  f"'{x)  =  6,  consequently,  in  this  case,  /^"(x)  and  all 
higher  derivatives  vanish. 

The  Geometrical  Meaning  of  the  Second  Derivative^ 

73.  If  the  curve  whose  equation  is 


74 


SUCCESSIVE  DIFFERENTIA  TION. 


[Art.  73. 


be  constructed,  we  have  seen  (Art.  26)  that 

^  being  the  inclination  of  the  curve  to  the  axis  of  x  \  hence 


/"W 


_    ^(tan  ^) 
~dx 


Fig.  6. 


If  now  the  value  of  this  derivative  be  positive, 
tan  ^  will  be  an  increasing  function  of  x,  as  in 
Fig.  6,  in  which,  as  we  proceed  toward  the 
right,  tan  ^  (at  first  negative)  increases  alge- 
braically throughout.  In  this  case,  therefore, 
the  curve  appears  concave  when  viewed  from 
above.  On  the  other  hand,  if  f"{x)  be  negative,  tan  ^  will  be 
a  decreasing  function  of  x,  as  in  Fig,  7,  in 
which,  as  we  proceed  toward  the  right,  tan  <i> 
decreases  algebraically  throughout,  the  curve 
appearing  convex  when  viewed  from  above. 


Fig.  7. 


74.  A  point  which  separates  a  concave  from 
a  convex  portion  of  a  curve  is  called  a  point  of 
inflexion,  or  2,  point  of  contrary  flexure. 

It  is  obvious  from  the  preceding  article  that,  at  a  point  of 
inflexion,  like  P  in  Fig.  8,  f"(x)  must  change 
ugn ;  hence  at  such  a  point,  the  value  of  this 
derivative  must  become  either  zero  or  infinity. 

75.  When  a  curve  is  described  by  a  moving 
point,  the  character  of  the  curvature  is  depen- 
dent upon  the  component  accelerations  of  the 
motion.     For,  if  we  put 

Vx  =  r,        or        dx  •=€  dt^ 
c  denoting  a  constant,  we  have 


Fig.  8. 


§  XI.]  THE   SECOND  DERIVA  TIVE.  ^% 

and  hence  /"W  =  ?-^=f- 

Whence  it  follows  that,  if  Vx  is  constant,  ay  and  f"{x)  have 
the  same  sign,  and  consequently  that  a  portion  of  a  curve 
which  is  concave  when  viewed  from  above  is  one  in  which  ciy  is 
positive  when  ax  is  zero. 


Successive  Differentials, 

76.  The  successive  differentials  of  a  function  of  x  involve 
the  successive  differentials  of  x  ;  thus,  if 

we  have  dy  =  ^x^dx, 

dy  =  6x(dxy-\-  3;rW, 
and  dy  =  6{dxy  +  iSx  dxd'x  +  sx'd'x. 

In  general,  if 

dy=/Xx)dx, 

dy=/"{x){dxy+/Xx)d''x, 

and  dy  =/"'{x)  {dxy+  s/"{x)dxd'x  +/'{^)d'x. 

Equicrescent    Variables. 

11,  A  variable  is  said  to  be  equicrescent  when  its  rate  is  con- 

dx 
stant ;  since  dt  in  the  expression  — -  is  assumed  to  be  constant, 

dt 

dx  is  also  constant,  when  x  is  equicrescent. 

In  expressing  the  differentials  of  a  function,  it  is  admissible 


76  SUCCESSIVE  DIFFERENTIATION.  [Art.   77. 

to  assume  the  independent  variable  to  be  equicrescent,  since 
the  differential  of  this  variable  is  arbitrary.  This  hypothesis 
greatly  simplifies  the  expressions  for  the  second  and  higher  dif- 
ferentials of  functions  of  x^  inasmuch  as  it  is  evidently  equiva- 
lent to  making  all  differentials  of  x  higher  than  the  first  vanish. 
Thus,  in  the  general  expressions  for  d'^y  and  d^y  given  in  the 
preceding  article,  all  the  terms  except  the  first  disappear,  and 
it  is  easy  to  see  that,  in  general,  we  shall  have 

when  X  is  equicrescent. 

78.  From  the  above  equation  we  derive 

dx""       -^    ^  ' 

The  expression  in  the  first  member  of  this  equation  is  the  usual 
symbol  for  the  n\h  derivative  of  y  regarded  as  a  function  of  x. 
The  n\h.  differential  which  occurs  in  this  symbol  is  always  un- 
derstood to  denote  the  value  which  this  differential  assumes 
when  the  variable  indicated  in  the  denominator  is  equicrescent. 

The  symbol  —-  is  frequently  used  to  denote  the  operation 
dx 

of  taking  the  derivative  with  reference  to  ;r,  and  similarly  the 

/  d\''  d" 

symbol  ( ^7- )  ,  or  — -- ,  is  used  to  denote  the  operation  of  tak- 
ing the  derivative  with  respect  to  x,  n  times  in  succession. 


Examples  XL 

V  I.  Find  the  second  derivative  of  sec  x,  and  distinguish  the  concave 
from  the  convex  portions  of  the  curve  y  ~  sec  x.  Also  show  that  the 
curve  y  =  log  x  is  everywhere  convex. 


§  XL]  EXAMPLES.  77 

y    2.  Find  the  points  of  inflexion  in  the  curve  j/  =  sin  x, 
y  3.  Find  the  point  of  inflexion  of  the  curve 

y  =z  2X^  —  ^^x"^  —  I  2Jt:  +   6. 

The  point  is  (J,  —  J). 

^  4.  Show  that  the  curve  y  =  tan  x  is  concave  when  y  is  positive,  and 
convex  when  y  is  negative. 

V  5.  Find  the  points  of  inflexion  of  the  curve 

y  =  X*  —  2X^  —  l2Jt:^  +  11^  +  24. 

The  points  are  (2,  —  2)  and  (—  i,  4). 

/     6.  If/W=i±|,findrW.  /V)  =  (7^^e. 

yl     7.  If/(^)  =-,  find/    (x).        f    {x)=-- ^^, . 

^    8.  If  j^  is  a  function  of  x  of  the  form 

Ax""  -h  Bx""^  +  •  •  •  +  Mx  +  Ny 

prove  that  -j^  =  i.  2.  3  •  •  •  ;2  ^. 

UrX 

^   9.  If/(^)  =  nfind/M^). 
v/  10.  If/  [x)  =  x'  log  (w^),  find/^^  (^). 

/    II.  If/  (x)  =  log  sin  jc,  find/'"  (:r). 

1^  12.  If/  {x)  =  sec  ^,  find/"  (x)  and/'"  (.:«:). 

/"  (^)  =  2  sec^ ^  —  sec  x,  and/"'  (^)  =  sec  x  tan  ^  (6  sec^:*:  —  i). 
/  13.  If/  (x)  =  tan  X,  find/"'  {x)  and/'"  (^). 

/'"  (jc)  =  6  sec^jc  —  4  sec-^,  and/'"  (;t)  =  8  tan  .a;  sec^^i;  (3  sec^^^c  —  i). 


•'(*) 

=  a'(losdri>". 

ri^)=i. 

/" 

,             2  cos  X 

^   '         sin'  ^  * 

78 

J       18. 


SUCCESSIVE  DIFFERENTIATION,  [Ex.    XI. 

If/  W  =  x%  find/"  (jc).       /"  (^)  =  ^'  (i  +  log  xf  +  :^-\ 


If^  =  £^,findg. 
If^=e-=^^,findg. 


Ifj;  =  log(£^  +  f-*),findg. 

If  J 


dx'~  °(£-+f-y 


I        ^    .d'y       .d'y 


V  19. 

V    20, 

V    23. 


If  _y  =  sin  '  jv,  find  -y^. 


d''y  _  gx  +  6.v' 


If^  =£"»',  find  g. 


^3 

-T=i  =  —  £^^°*  COS  ^  sin  X  (sin  jc  +  3). 
dx 


li  y  = 


,  find     -^ 


d'^y  __      I  —  logjc 
^JC"*        ^  (i  +  log^)^ 


I  +  log  X  *  ^/jJC^ 

Find  the  value  of  ^^(g""),  when  x  is  not  equicrescent. 

d\e')  =z  e'{dxy  +  2,^'  d'x  dx  +  £'  d'x. 

73 
Find  the  value  of  -z-^  (sin  6),  Q  being  a  function  of  /. 


df' 


,  .      .  /doY  .        do     d'o  d^O 

(sine)  =  -  cose  (^-j  _  3  sme  -  •  ^  +  cose  — .. 


CHAPTER   V. 
The  Evaluation  of  Indeterminate  Forms. 


XII. 
Indeterminate   or   Illusory   Forms. 

79.  When  a  function  is  expressed  in  the  form  of  a  fraction 
each  of  whose  terms  is  variable,  it  may  happen  that,  for  a  cer- 
tain value  of  the  independent  variable,  both  terms  reduce  to 

zero.     The  function  then  takes  the  form  - ,  and   is  said  to  be 

.o 

indeterminate,  since  its  value  cannot  be  ascertained  by  the  ordi- 
nary process  of  dividing  the  value  of  the  numerator  by  that 
of  the  denominator.  The  function  has,  nevertheless,  a  value  as 
determinate  for  this  as  for  any  other  value  of  the  independent 
variable.  It  is  the  object  of  this  chapter  to  show  that  such  defi- 
nite values  exist,  and  to  explain  the  methods  by  which  they 
are  determined. 

The  term  illusory  form-^s  often  used  as  synonymous  with 
indeterminate  form,  and  these  terms  are  applied  indifferently, 

not  only  to  the  form  -  ,  but  also  to  the  forms  — ,  co-  o,  co  —  oo, 

O  00 

and  to  certain  others  whose  logarithms  assume  the  form  oo-o. 
When  a  function  of  x  takes  an  illusory  form  for  x—a,  the  cor- 
responding value  of  the  function  is  sometimes  called  its  limits 
ing  value  as  x  approaches  the  value  a. 

80.  The  values  of  functions  which  assume  illusory  forms  may 


So  EVALUATION  OF  INDETERMINATE  FORMS.  [Art.  8o. 

sometimes  be  ascertained  by  making  use  of  certain  algebraic 
transformations.     Thus,  for  example,  the  function 

a  —  V{a^  -  bx) 

X 

takes  the  form  -  when  x  =  o. 
o 

Multiplying  both  terms  by  the  complementary  surd 

a  +  V{a^  —  bx\ 

bx  b 


we  obtain 


x\a  +  ^/{d'  -  bx)]      a  +  V(«'  -  bx)  ' 


The  last  form  is  not  illusory  for  the  given  value  of  x,  since  the 
factor  which  becomes  zero  has  been  removed  from  both  terms 
of  the  fraction.      The  value  of  the  fraction  for  x  =  o  is  evi- 
dently — . 
2a 

The  following  notation  is  used  to  indicate  this  and  similar 
results ;  viz., 

a  -  V{a'  -  bx)-]  _  b 


']: 


2a 


the  subscript  denoting  that  value  of  the  independent  variable 
for  which  the  function  is  evaluated. 


Evaluation   by   Differentiation, 

81.   Let  -  represent  a  function  in  which  both  u  and  v  are 
u 

functions  of  ;tr,  which  vanish  when  x  =■  a\  in  other  words,  for 

this  value  of  x,  we  have  u  =  Of  and  v  =  o. 


§  X 1 1 .]  EVAL  UA  TION  B  Y  DIFFERENTIA  TION, 


8i 


Let  P  be  a  moving  point  of  which  the  abscissa  and  ordinate 
are  simultaneous  values  of  u  and  v  {x  not 
being  represented  in  the  figure) ;  then,  de- 
noting the  angle  POU  hy  6,  and  the  inclina- 
tion of  the  motion  of  P  to  the  axis  of  u  by  ^, 
we  have 

dv 


Fig.  9. 


tan  (9  = 


and 


tan0  = 


du 


At  the  instant  when  x  passes  through  the  value  a,  u  and  v 
being  zero  by  the  hypothesis,  P  passes  through  the  origin  ;  the 
corresponding  value  of  6  is  evidently  determined  by  the  direc- 
tion in  which  P  is  moving  at  that  instant,  and  is  therefore  equal 
to  the  value  of  (j)  at  that  point. 

Hence  the  values  of  tan  6  and  tan  (j)  corresponding  to  ;ir  =  ^ 
are  equal,  or 


therefore,  to  determine  the  value  of  -  for  ;ir  =  a:  we  substitute 

21 

for  it  the  function  -7— ,  whose  value  is  the  same  as  that  of  the 
du 

given  function,  when  x  =  a. 


82.  This  result  may  also  be  expressed  in  the  following  man- 
ner :  let  f(x)  and  (l>{x)  be  two  functions,  such  that  f{d)  =  o, 
and  (l>(a)  =  o  ;  then 

<l>{a)      <l>\d) 


(I) 


As  an  illustration,  let  us  take 


log^ 


When x=i,  this  func- 


tion takes  the  form  — ;  by  the  above  process,  we  have 


82  EVALUATION  OF  INDETERMINATE  FORMS.    [Art.  82. 

log^n     ^n 
^-iJi      I  Ji 

the  required  value. 

dv         fix) 
83.  Since  the  substituted  function  -^  or  -^-rr—i    frequently 

du         (p  {x) 

takes  the  indeterminate  form,  several  repetitions  of  the  process 

are  sometimes  requisite  before  the  value  of  the  function  can  be 

ascertained. 

For  example,  the  function — takes  the  form  -  when 

tf  o 

6  =  o\  employing  the  process  for  evaluating,  we  have 


—  cos 


_  sin  8~\ 


v/hich  is  likewise  indeterminate ;  but,  by  repeating  the  process, 
we  obtain 


-  cos  ^"1  _  sin  ff 


cos  6~ 


=  h 


84.  If  the  given  function,  or  any  of  the  substituted  func- 
tions, contains  a  factor  which  does  not  take  the  indeterminate 
form,  this  factor  may  be  evaluated  at  once,  as  in  the  following 
example. 

The  function 

(l  —  ;ir)  f^  —  I 
tan''  X 

is  indeterminate  for  x  =  O.     By  employing  the  usual  process 
once,  we  obtain 

(l  -  X)8^  —  l"|    _  —X6^  "I 

tan'  X       J  o~    2  sec';ir  tan  xj  J 

which  is  likewise  indeterminate  ;  but,  before  repeating  the  pro- 

cess,  we  may  evaluate  the  factor -^—     .      The   value  of 

2  sec  X  |q 

this  factor  is  —  J  ;  hence  we  write 


§  XiL]  ABBREVIATED   METHODS.  83 


a=- 


(l  —  ;ir)  f-^  —  i-|  _  _  x^^ 

tan'  X 


sec'  X 


85.  When  the  given  function  can  be  decomposed  into  fac- 
tors each  of  which  takes  the  indeterminate  form,  these  factors 
may  be  evaluated  separately.     Thus,  if  the  given  function  be 

(f-^  —  i)  tan';t: 


may  be  employed.     We  have 
tan  X 


I  =  I,    and    — !■  I  =  I ; 

-Jo  X        _lo 


hence  the  value  of  the  given  function  is  unity. 

When  this  method  is  used,  if  one  of  the  factors  is  found 
to  take  the  value  zero  while  another  is  infinite,  their  product, 
being  of  the  form  o  •  00,  must  be  treated  by  the  usual  method, 
since  o  •  00  is  itself  an  illusory  form. 

86.  Another  mode  of  decomposing  a  given  function  is  that 
of  separating  it  into  ^^arts,  and  substituting  the  values  of  such 
parts  as  are  found  on  evaluation  to  be  finite. 
As  an  illustration,  we  take  the  expression, 


_  (g-^  -  8-^y-  2x\e^  +  8-^) 


1- 


Each  of  the  fractions  into  which  this  function  can  be  decom- 
posed being  obviously  infinite,  we  'first  apply  the  usual  process, 
thus  obtaining 


84  EVALUATION  OF  INDETERMINATE  FORMS.  [Art.  Z6, 


fi,Q        —  —  -  ^- 

4^ 


Separating  this  expression  into  two  fractions,  thus, — 

(f-r  4.  ^-r)  (f.r  __  g-r  _  2x)~\  6^  —  g-^n 

Jo  2X       Jo  ' 


Uo  = 

2X^ 


the  latter  is  found  on  evaluation  to  have  a  finite  value,  and  the 
expression  reduces  to 


e^  —  8-^  —  2x~] 

"°  =  — p — J„-  '• 

Hence 

"»  ==  — IP— 1- '  = -6^1- '  =  -  *• 

Examples  XI L 

/        ^          sin  x~\                tan  ^"1                      ,      f'  —  i"l 
V    I.  Prove  =  I,      =  I,      and      = 

X     _Jo  X      |o  X       |o 


I. 


These  results  are  frequently   useful  in   evaluating   other  functions. 
Evaluate  the  following  functions  : 

V  6*  —  f 

2. -. , ,  when  x^^  o.  2. 

log(i+^) 

/  or  -  x"" 

log  ^  —  log  a; 


J         ■y'-5-^'+  7-^-3 


4-    ^»_>_5^_3  »  -^-3.  ^ 

/  X*  —  d>x^  +  22:1:^  —  24JC  +  9  _  I 

^*    ^*  —  4^"  —  2Jt:'  +  12^  +  9  '  -^  —  3-  - 


/    «  £f-r 


> 


6**—  I 


:r  =  o.  —  I . 


§  XII.]  EXAMPLES.  85 

f  sin  ^  —  cos  X 


is. 


,  when  X  =  In,         I V2. 

sin  2X  —  cos  2X  —  I 


log  JC 


V(i  —  X) 


o. 


s/     9-  -^— »  ^  =  °-  log-. 

V     10.  — ^ —-^ -,  (See  Art.  84),  x  =  1.  —  . 

\r  V  „.  ?i^"-(x  -  cos  ^), 


X  ^  o. 


V'      12.  ,  x^=a.  wf" 


.;c  —  ^ 


/    13.  :j -. ,  r.  —  \7t.      aloga. 

/  I   —  COSX'  I 

V      14-  ~i 7 r>  .r;  =  o.  — . 

*  ^     X\0%{\    -\-  XY  2 

i/:r  tan  x 

15.   -^,  x=o.  I. 

/•?//  /«  the  form    i/ • .     See  Art.  2><  and  Ex- 

ample  i. 


Vx  —  Va  -V-  |/(-^  —a)  _  I 


JC^/C^JC  —   2JC*)  —  X^  81 

I     17.  ^^^ r y  x=i,               — 

^        '                 1-x^  .20 

/      ^    (d"  +  ax^-  x"")^  -  (a"  -  ax  ■\-  x^)^ 

^/     18.^ , \ -T ^,  x  =  o.               Va 

{a  +  xy^  —  {a  —  xY 

Multiply  both  terms  by  the  two  complementary  surds.     See  Art.  80. 


86  EVALUATION  OF  INDETERMINATE  FORMS.  [Ex.  XI I. 

•   \\  10.  -^ -, ^^ n  ,  when  x-=-a.    — ; -r  . 

Divide  both  terms  by  (a  —  x)^. 


smx  —  X  cos  ^ 

^        20. 


V    24. 


X  —  smx      * 

«* 

-  €~'  -  2X 

Jt^— tanjc      ' 

{x-  2)e'  +  X  -\- 

2 

xia'-iy 

x"  —  X 

I 

—  X  -{-  log  X* 

tan  x  —  smx 

Sin  X  1     sec  ^  "~~  i  I 
jPut  in  the  form    • 5 . 

^     Jo  X  _\o 

J  (^  —  i)^  +  sin^(jc''  —  i)^ 

^^'  (^  +  i)  (^  -  i)*       ' 


28. 


^TT  —  tan~'  ^ 

^n   ^8in(logx)     f 


X  =  O.  2. 


^=  O. 


V/        21. 


;v  =  o. 


^=1.  '^2. 


I  —    V(2X  —  X) 

t          sin^jc  —  log(6''cos^) 
yj  27.  -^ , 


I 

a:  =  0. 

2' 

I 

X  —  I. 

2(l-«)- 

x  =  o.      (I 

4-d!')sec'«. 

i  tan  (^  +  -y)  ~  tan  (d?  —  x) 

V   ^9'  tan-'  (a  +  x)  —  tan"'  (a  -  ^)' 

J         xsinx  —  irr  ^  —  1  ^ 

>/  -o.  ,  x  =  i7r.         —I. 

^  cos  X 


§  XII.]  EXAMPLES.  87 

fx  c  8in  a 

\i   %\.   -. — ,  when  ^  =  o.  i. 

nf  sin  nx  —  if  sin  mx 

J    -12.  ,  m  =  n. 

^    ^         tan  ?tx  —  tan  mx 

if~^{n  cos  nx  —  sin  nx^  cos''  nx. 

Ifi  solving  this  and  the  follcnmng  example,  x  and  n  inay  be  regarded 
as  co7istants,  a?id  m  as  a  variable. 

ytan  nx  —  tan  mx  sec"  nx 

^^     sm  {nx  —  mx)  2« 


XIII. 

TAe  Form  -^. 

f(x) 
87.  Let  ^;^  denote  a  function  which  assumes  the  form 
(p(x) 

-^  when  x=.  a,  then  we  have 

I 

(I) 

The  second  member  of  this  equation  takes  the  form  -  when 

o 
x  =  a\  we  therefore  have,  by  equation  (i)  Art.  82, 

I  ^'(^) 


/w_ 

.-K^) 

4>{x) 

I 

f{a)  _  ({(^)  _       [(^(^)]'_  <t>\a)  \Ad)  y 

whence,  if  •^;  /   is  neither  zero  nor  infinity,  we  infer  that 
(pia) 


88  EVALUATION  OF  INDETERMINATE  FORMS.  [Art.  87. 

<t>{a)- 4>' ia) ^^^ 

This  formula,  it  will  be  observed,  is  identical  with  that  employed 

when  the  function  takes  the  form  — . 

o 

88.  When  the  value  of  "44-^  is  either  zero  or"  infinity,  equa- 

tion  (2),  Art.  87,  will  be  satisfied  independently  of  the  exist- 
ence of  equation  (3) ;  we  are  not  justified  therefore,  when  this 
is  the  case,  in  deriving  the  latter  from  the  former.  The  follow- 
ing demonstration  shows,  however,  that  equation  (3)  holds  in 
these  cases  also. 

First,  when   the  value  of   4;  x   ^^  zero,  by  adding  a  finite 
quantity  n  to  the  given  function,  we  have 

a  function  which  is  by  hypothesis  finite.  To  this  function  there- 
fore the  demonstration  given  in  Art.   87  applies ;  hence 

therefore  -^  =-^ 

tneretore  ^^^^      ^.^^y 

as  before. 

Again,  if  the  value  of  v^-t  is  infinite,  that  of  ^^r~  is  zero, 

<f>{a)  /(a) 

and,  by  the  last  result, 

4  {a)  _  <l>'ia) 
f{a)-f{a)' 

hence,  in  this  case,  likewise 


§  XIII.]  THE  FORM  f .  89 

f{d)_f\a) 


Derivatives  of  Functions  which  assume  an  Infinite  Value, 

89.  When  f(x)  becomes  infinite,  for  a  finite  value  a  of  the  in- 
dependent variable,  f  '(a)  is  likewise  infinite.  For,  let  b  denote  a 
value  of  X  so  taken  that  fix)  shall  be  finite  iov  x  =^  b  and  for  all 
values  of  x  between  b  and  a  :  then,  as  x  varies  from  b  to  a,  the 
rate  of /(;ir)  must  assume  an  infinite  value,  otherwise /"(jtr)  would 
remain  finite.  The  value  of  x  for  which  the  rate  is  infinite  must 
be  a  or  some  value  of  x  between  b  and  a ;  that  is,  some  value 
of  X  nearer  to  a  than  b  is.  Now,  since  b  may  be  taken  as  near 
as  we  please  to  a,  the  value  of  x  for  which  the  rate  is  infinite 

dx 
cannot  differ  from  a.     The  expression  for  this  rate  is/"(4r)  — -,  in 

which  -^  may  be  assumed  finite,  therefore  f'{x)  must  be  infinite 
when  X  =  a;  in  other  words,  f'(a)  is  infinite  when  f(a)  is  infinite. 

90.  It  follows  from  the  theorem  proved  in  the  preceding 
article  that  when  a  is  finite  the  function  obtained  by  the  appli- 
cation of  formula  (3),  Art.   87,  takes  the  same  form,  —  ,as  that 

00 

assumed  by  the  original  function.  Hence,  except  when  the 
given  value  of  x  is  infinite,  the  application  of  some  other  process, 
either  to  the  original  function  or  to  one  of  the  substituted  func- 
tions, is  always  requisite.     Thus  in  the  example, 

log  (sin  2xy^  _  00  ^ 
log  sin  ;r  Jo      00  * 

by  using  the  above  formula  we  obtain 


90  EVALUATION  OF  INDETERMINATE  FORMS.  [Art.  90. 

log  sin  2x~\  _  2  cot  2x' 


G 


log  sin  ^  Jo        cot  X 


00 


which  takes  the  form  — ;  but  the  last  expression  is  equivalent 


00 


sin  X  cos  "Zx  I 

to  2  — ,  and  is  therefore  easily  shown  to  have  the 

sm  2x  cos  xAo 

value  unity. 


The  Form  o .  oo. 

91.  A  function  which  takes  this  form  may,  by  introducing 
the  reciprocal  of  one  of  the  factors,  be  so  transformed  as  to  take 

either  of  the  forms  -  or  _ ,  as  may  be  found  most  convenient. 

o         00 
For  example,  let  us  take  the  function 

which  assumes  the  above  form  when  x  =  ^^n  being  positive. 
In  this  case  it  is  necessary  to  reduce  to  the  form  — .     Thus — 

x-n^  —  -^\    = =  _ ^ ,  etc. 

By  continuing  this  process,  we  finally  obtain  a  fraction  whose 
denominator  is  finite  while  its  numerator  is  still  infinite.  Hence 
we  have,  for  all  finite  values  of  n, 

X-'^  £^1     =  00. 
J  00 

The  Form  oo  —  oo. 

92.  A  function  which  assumes  this  form  may  be  so  trans- 
formed as  to  take  the  form  - .     Let  the  given  function  be 


§  XIII.]                                  THE  FORM   00-00.  Ql 

r       J log  (I  +-^)"] 

which  takes  the  form  oo—(x,  since  the  second  term  is  easily- 
shown  to  be  infinite.     But 

r        I         _  log  (I  +-^)1  _;ir~(i  +.y)log(l  4-.r)"]  ^  " 
Lr(i  +  X)  x'        Jo  ;rXi  +  x)  Jo 

_;r-(l  +;r)log(l  +  .y)"! 

Jo  2 


;i;' 


I  —  log  (i  4-  x)  —  i 

2X 


Examples  XIII. 
Evaluate  the  following  functions  : 


sec^ 
sec  ^x 


a' 


\l     2.- 

cosec  (ma  ')  * 
V    ^.  i^^ 


when  X  =  ^zr.  —  3. 


X  =  CO.  m. 


{n  >  o),  x=.  00. 


/. 


tan  jc 


log  {x  —  \7t)  ' 


/  sec  (JTTjc) 

^  5-    log(i_^)' 

/  g    log  cos  (iTTjy) 

^  •     log(i-^)   ' 


X  =  J;r.  00. 


.ijf  =  I.  00. 

JP  =  I.  I. 


92  EVALUATION  OF  INDETERMINATE  FORMS.  [Ex.  XIII. 

tan^ 

log(i  +:r) 

/      <>• ^^ »  JC  =   00.  o. 

^    9-    (^^'  —  ly^,  a:  =^00.  log ^. 


TtX 


>! 


X 

I 


13.    £     -(l  —  logx), 


log  tan  Jt  ' 


log  cot  — 


JIC  =  o. 


COtJt'  +  log^' 
>)  17.  SQCx{xsmx  —  ^rc),  x  =  ^7i:. 


X  =  a. 


A  i8.  log  (2  -  - )  tan  —  , 

\)  19.  (i  —  ^)  tan  (^7rx)y  x  =  1 

J    20.  log  {x  —  a)  tan  (^  —  a). 


X  =a. 


y 


4 


V    10.  ^ —  tan  —  ,  X  =  a 

II.  x'*{\ogxY\  (m  and  Xi  being  positive)^  x  =  o.  o. 

^    12.  £^sin-,  ^=00.  00. 


/                     TtX     ^         1  2 

V   14.  sec log-,  x=i.                — . 

2            °  X  Tt 

J        logtan;zx 

V  15.  _^_ ^  ^  =  0.                  I. 


—  I. 


§  XIV.]  THE  FORMS  c»%  o\  AND  i  °°.  93 

XIV. 

Functions  whose  Logarithms  take  the  Form  oo .  o. 

93.  In  the  case  of  a  function  of  the  form  u"^  we  have 

log  uy=v  log  u. 

The  expression  v  log  u  takes  the  illusory  form  o  •  oo  in  two 
cases :  first,  when  v  —  o  and  log  u  —  on  \  and  secondly,  when 
v=  CO  and  log  u  —  O. 

Log  ?/  is  infinite  when  u^o,  and  also  when  u=  co]  there- 
fore the  first  case  will  arise  when  the  original  function  takes 
one  of  the  forms    00°  or  0°. 

'Logu  =  o  when  u=  i,  therefore  the  second  case  will  arise 
when  the  original  function  takes  the  form  i  °°. 

Hence  functions  which  take  either  of  the  three  illusory  forms. 

00°,         0°,         or         1% 

may  be  evaluated  by  first  evaluating  their  logarithms,  which 
take  the  form  o  •  co. 

It  is  to  be  noticed  however  that  0°°  and  00°°  are  not  illu- 
sory forms,  since  their  logarithms  take  the  form  00  (ip  00). 


The  Forfn    i  °". 
94.  As  an  illustration  of  this  form,  we  take  the  function 
which  assumes  the  form  i  °°  when  ;ir  =  00.     Denot- 
ing this  function  by  u,  we  have 


(-1) 


log2^  =  ;irlog/^l  -h  -j 


the  last  expression  assuming  the  form  -  when  x  —  (j^. 


94  EVALUATION    OF  INDETERMINATE  FORMS.  [Art.  94. 

In  evaluating  this  logarithm,  it  is  convenient  to  substitute 
z  ior  —  \  then 

,  log(i4-^^)"l 

since,  when  ;tr  =  00,  z  —  o.     Taking  derivatives,  we  have 

Z  Jo  I   +  ^^Jo 

Hence  u^=\\-\--\        =  a*. 

95.  If  <2:  =  I,  we  have 

that  is,  as  x  increases  indefinitely,  the  limiting  value  of  the  func- 
tion (  I  H — j    is  f.     The  Napierian  base  is  often  defined  as  the 

limiting  value  of  this  function,  or,  what  is  the  same  thing,  by- 
formula 

f=(i  ^x)i\^. 

The  Form   o°. 

96.  The  function  jtr^j^,  by  the  aid  of  which  many  functions 
of  similar  form  may  be  evaluated,  will  serve  as  an  illustration 
of  the  form  0°. 

Let  u  =  x^\ 

then  logu  =  x\ogx, 


§  XIV.]  THE  FORM  o\  95 

and  logu]  =  -^-^     =  —  -^     =  o; 

_Jo  -^  _lo  -^       -Jo 


therefore  x^\ 


I. 


The  value  of  a  function  which  takes  the  form  o°  is  usually 
found,  as  in  the  above  example,  to  be  unity.  This  is  not,  how- 
ever, universally  true,  as  the  function 

a  +  x 

(one  of  those  earliest  adduced  for  this  purpose  *)  will  show. 

This  function  takes  the  form  o°,  when  x  —  0\  but  since  its 
logarithm  reduces  to  a  -\-  x,  its  value  when  ;r  =  o  is  £^. 

Examples  XIV. 

>/ I.  (cos  ^)'=°^'-^,  when;t:  =  o.  e~* 

J  /tan^Xja 

/  -- 

^    3.    (cosa'Jtr)^oseca/j.r^  ^  =  O.  6^^'. 

—        5.  (tan^m  :^=-i7r.  i.       £, 


nI    6.   ('^„')"(^>o). 


Kx"" 


o.  I. 


vl     7.  (i-^)%  *  =  o.  ^. 

4    8.  (sinjc)««^'^  ar  =  j7r.  £-*. 


*  See  Crelles  Journal,  vol.  xii,  p.  293. 


96  EVALUATION  OF  INDETERMINATE  FORMS,   [Ex.  XIV. 

Solution:     (cot ^r^]^=  [^-|&°  =  i.     (5..  ^r/.  96.) 
^'   10.  (sin  xf^',  x  =  o.  .1. 


V    20.  (i  ±  ^)^, 
/21.  ;r"(sin^)*^°Y'^~^-^V, 

\2  Sm  2JC/ 

{m'  —  i)  (asinx  —  sin  dJJi;) 


22. 


_  ^ 
2 


■>1   II.  (sinjt:)'^"', 

a 
v/l2.    ^log^i^, 

^   13.  (sinj^y^et^-^,  ^=0.  e'**. 


x  =  o. 


V  14.  ^^  (^  >  o), 

V  15.    (^')log(-^  +  logcosjr)^ 

V  — 

16.  :x:^   -^j  when  ^  =  i. 

V  17.  •^'"S 


:^  =  o.  I. 


X  =  o.  f2«'. 


.^t:  =  o. 


^    18.   (cos  ^zj*;)-^",  ^  =  0.  f-i«^2 

/:..('^)t 


;r  =:  00. 


.:r  =  00. 


n 
X  =  — . 
2 


,m+3 


^-^  sin  ^  (cos  X  -  cos  ^^)»  '  ""  -  °'   (  3  ]  ^^S"^- 


V   / 


CHAPTER   VI. 

Maxima  and  Minima  of  Functions  of  a  Single 
Variable. 


XV. 

Conditions  Indicating  the  Existence  of  Maxima 
and  Minima, 

97.  If,  while  the  independent  variable  increases  continu- 
ously>  a  function  dependent  on  it  increases  up  to  a  certain 
value,  and  then  decreases,  this  value  of  the  function  is  said  to 
be  a  maximum  value.  In  other  words,  a  function  f{x)  has  a 
maximum  value  corresponding  to  ;r  =  ^,  if,  when  x  increases 
through  the  value  ^,  the  function  changes  from  an  increasing 
to  a  decreasing  function. 

Since  f'{x)  is  positive,  when  f{x)  is  an  increasing  function, 
and  negative  when  it  is  a  decreasing  function ;  it  is  obvious 
that  li  f(d)  is  a  maximum  value  of  f{x)yf'{x)  must  change  sigUj 
from  +  to  — ,  as  ;r  increases  through  the  value  a. 

On  the  other  hand,  a  function  is  said  to  have  a  minhnum 
value  for  x=  a^  if  it  is  a  decreasing  function  before  x  reaches 
this  value  and  an  increasing  one  afterward.  In  this  case,  f'(x) 
changes  sign  from  —  to  +. 

98.  The  derivative  f'{x)  can  only  change  sign  on  passing 
through  zero  or  infinity.  Hence  a  value  of  x,  for  which  f{x) 
is  a  maximum  or  a  minimum^  must  satisfy  one  of  the  two  follow- 
ing equations  : 

f\x)  =  o  and  /'{x)  =  oo. 


98  MAXIMA    AND  MINIMA.  [Art.  98. 

The  required  values  of  x  will  therefore  be  found  among  the 
roots  of  these  equations. 

The  case  which  usually  presents  itself,  and  which  will  there- 
fore be  considered  first,  is  that  in  which  the  required  value  of 
;r  is  a  root  of  the  equation  f\x)  =  o. 

99.  As  an  illustration,  let  it  be  required  to  divide  a  number 
into  two  such  parts  that  the  square  of  one  part  multiplied  by  the 
cube  of  the  other  shall  give  the  greatest  possible  product. 

Denote  the  given  number  by  ^,  and  the  part  to  be  squared 
by  X  ;  then  we  have 

f(x)  =  x\a-x)\ 

It  is  evident  that  a  maximum  value  of  this  function  exists ; 
for  when  ;ir  =  o  its  value  is  zero,  and  when  x  =  a  its  value  is 
again  zero,  while  for  intermediate  values  of  x  it  is  positive ; 
hence  the  function  must  change  from  an  increasing  to  a  decreas- 
ing function  at  least  once,  while  x  passes  from  the  value  zero  to 
the  value  a. 

Taking  the  derivative  of  this  function,  the  equation 

is  in  this  case       2x{a  —  xy  —  34:^^  {a  —  x^  =  o, 

or  x{a  —  xy  {2a  —  $x)  =  o. 

o  and  a  are  roots  of  this  equation ;  but,  as  we  are  in  search  of 
a  value  of  the  function  corresponding  to  an  intermediate  value 
of  Xy  we  put 

2a  —  ^x  =  o, 

and  obtain  x  =  ^a. 

The  corresponding  value  of  the  function  is  -^^a^y  the  maxi- 
mum value  sought. 


§XV.] 


GEO  ME  TRIG  A  L  MA  GNI  TUBES. 


99 


Maxima  and  Minima  of  Geometrical  Magnitudes, 

100.  When  the  maximum  or  minimum  value  of  a  geometri- 
cal magnitude  limited  by  certain  conditions  is  required,  it  is 
necessary  to  obtain  an  expression  for  the  magnitude  in  terms  of 
a  single  unknown  quantity,  such  that  the  determination  of  the 
value  of  this  quantity  will  constitute  the  solution  of  the  prob- 
lem. For  example  :  let  it  be  required  to  deterinine  the  cone  of 
greatest  convex  surface  among  those  which  can  be  inscribed  in  a 
sphere  whose  radius  is  a. 

Any  point  A  of  the  surface  of  the 
sphere  being  taken  as  the  apex  of 
the  cone,  let  the  diagram  represent 
a  great  circle  of  the  sphere  passing 
through  the  fixed  point  A, 

If  we  refer  the  position  of  the 
point  P  to  rectangular  coordinates, 
and  take  C  as  the  origin,  the  required 
cone  will  evidently  be  determined 
when  X  is  determined.  We  have 
now  to  express  the  convex  surface 
vS  in  terms  of  x. 
The  expression  for  the  convex  surface  of  a  cone  gives 

S=ny^/lf^{a^xy\      .....     (l) 

in  which  the  unknown  quantities  x  and  y  are  connected  by  the 
equation  of  the  circle 

X-'^f^d^ (2) 

Substituting  the  value  of  j/,  we  have 

5  =  TT  4/(^'  -  ^0  4/(2^'  +  2ax\ 
reducing,  S=7t  V{2a)  {a  +  x)V{a—x) (3) 


Fig.  10. 


lOO 


MAXIMA    AND  MINIMA. 


[Art.  lOO. 


Since  the  factor  n  V(2a)  is  constant,  we  are  evidently  re- 
quired to  find  the  value  of  x  for  which  the  function 

/{x)  =z{a  -i-  x)  Via  -  x) 

is  a  maximum.     The  equation  /\x)  =  o  is,  in  this  case, 

a  -\-  X 


V{a  -  x) 


whence 


2  V{a  —  x) 
x  =  \a. 


=  o; 


The  altitude  of  the  required  cone  is  therefore  \a.     Substi- 
tuting this  value  of  x  in  equation  (3),  we  have 

S=^Vy7ta\ 

the  maximum  value  required. 

fOI.  As  a  further  illustration,  let  it  be  required  to  determine 
the  greatest  cylinder  that  can  be  in- 
scribed in  a  given  segment  of  a  pa- 
raboloid of  revolution. 

Let  a  denote  the  altitude,  and  b 
the  radius  of  the  base  of  the  seg- 
ment.    The  equation  of  the  gener-   ^f x_ 

ating  parabola  is  of  the  form 


V   =  A^cx. 

Since  (^,  b)  is  a  point  of  the  curve, 
we  have  the  condition 

b"^  =  4ca  ; 
eliminating  4^,  the  equation  of  the  curve  is 

b' 


Fig    II. 


y  =  -  ;ir. 
a 


(I) 


§  XV.]  GEOMETRICAL   MAGNITUDES.  10 1 

The  volume   V  of  the  cylinder  of  which  the  maximum  is  re- 
quired is  expressed  by 

V  —  ny'ia  —  x)^ 
or,  by  equation  (i),  V  —  n  —  x{a  —  x). 

Hence  we  put  fix)  —  ax  —  x\ 

and  the  condition  /'{x)  =  o  gives 

X  =  ^a. 

Consequently  a  —  Xy  the  altitude  of  the  cylinder,  is  one  half  the 
altitude  of  the  segment. 

Examples    XV. 

y    I.  Find  the  sides  of  the  largest  rectangle  that  can  be  inscribed  in 
a  semicircle  of  radius  a.  The  sides  are  a  V2  and  \a  V2. 

yj  2.  Determine  the  maximum  right  cone  inscribed  in  a  given  sphere. 
The  altitude  is  four  thirds  the  radius  of  the  sphere. 

y  3.  Determine  the  maximum  rectangle  inscribed  in  a  given  segment 
of  a  parabola. 

The  altitude  of  the  rectangle  is  two  thirds  that  of  the  segment. 


J 


4.  Find  the  maximum  cone  of  given  slant  height  a. 

The  radius  of  the  base  is  \a  V6. 


\^  5.  A  boatman  3  miles  out  at  sea  wishes  to  reach  in  the  shortest 
time  possible  a  point  on  the  beach  5  miles  from  the  nearest  point  of 
the  shore  ;  he  can  pull  at  the  rate  of  4  miles  an  hour,  but  can  walk  at 
the  rate  of  5  miles  an  hour  ;  find  the  point  at  which  he  must  land. 

Express  the  7vhoh  time  hi  terms  of  the  distance  of  the  required  point 
from  the  nearest  point  of  the  shore. 

He  must  land  one  mile  from  the  point  to  be  reached. 


102  MAXIMA   AND  MINIMA.  [Ex.  XV. 

V  6.  If  a  square  piece  of  sheet-lead  whose  side  is  a  have  a  square  cut 
out  at  each  corner,  find  the  side  of  the  latter  square  in  order  that  the 
remainder  may  form  a  vessel  of  maximum  capacity 

The  side  of  the  square  is  \a. 

%/  7.  A  given  weight  is  to  be  raised  by  means  of  a  lever  weighing  n 
pounds  per  linear  inch,  which  has  its  fulcrum  at  one  end,  and  at  a 
fixed  distance  a  from  the  point  of  suspension  of  the  weight  w  ;  find  the 
length  of  the  lever  m  order  that  the  power  required  to  raise  the  weight 
may  be  a  minimum.  /2a'w 

J  .  ^^ 

V  8.  A  rectangular  court  is  to  be  built  so  as  to  contain  a  given  area 
<:',  and  a  wall  already  constructed  is  available  for  one  of  the  sides  ; 
find  its  dimensions  so  that  the  least  expense  may  be  incurred. 

The  side  parallel  to  the  wall  is  double  each  of  the  others. 


^ 


J 


9.  Determine  the  maximum  cylinder  inscribed  in  a  given  cone. 
The  altitude  of  the  cylinder  is  one  third  that  of  the  cone. 


10.  Prove  that  the  rectangle  with  given  perimeter  and  maximum 
area  is  a  square  ,  also  that  the  rectangle  with  given  area  and  minimum 
perimeter  is  a  square. 


i 


II.  Find  the  side  of  the  smallest  square  that  can  be  inscribed  m  a 
square  whose  side  is  a. 

Take  as  the  independent  variable  the  distance  between  the  angles  of  the 
two  squares.  ia  ^2. 


^ 


12    Inscribe  the  maximum  cone  in  a  given  paraboloid,  the  apex  of 
the  cone  being  at  the  middle  point  of  the  base  of  the  paraboloid. 

The  altitude  of  the  cone  is  half  that  of  the  paraboloid. 


13.  Find  the  maximum  cylinder  that  can  be  inscribed  in  a  sphere 
whose  radius  is  a.  The  altitude  is  ^a  V3. 

s^  14.  Through  a  point  whose  rectangular  coordinates  are  a  and  b  draw 
a  line  such  that  the  triangle  formed  by  this  line  and  the  coordinate 
axes  shall  be  a  minimum. 

The  intercepts  on  the  axes  are  2a  and  2d, 


§  XV.]  EXAMPLES.  103 


y 


/ 


15.  A  high  vertical  wall  is  to  be  braced  by  a  beam  which  must  pass 
over  a  parallel  wall  a  feet  high  and  b  feet  distant  from  the  other , 
find,  the  length  of  the  shortest  beam  that  can  be  used  for  this  purpose. 

Take  as  the  independent  variable  the  inclination  of  the  beam  to  the 
horizon 

{J^  +  P)  . 

16.  The  illumination  of  a  plane  surface  by  a  luminous  point  being 
directly  as  the  cosine  of  the  angle  of  incidence  of  the  rays,  and  in- 
versely as  the  square  of  its  distance  from  the  point ;  find  the  height 
at  which  a  bracket-burner  must  be  placed,  in  order  that  a  point  on 
the  floor  of  a  room  ^t  the  horizontal  distance  a  from  the  burner  may 
jeceive  the  greatest  possible  amount  of  illumination. 

The  height  is  —y. 


XVI. 


Methods  of  Discriminating  between  Maxima  and  , 

Minima, 

102.  When  the  existence  of  a  maximum  or  a  minimum  cor- 
responding to  a  particular  root  a  of  the  equation  f\x)  =  o  is 
not  obvious  from  the  nature  of  the  problem,  it  is  necessary  to 
determine  whether  f\x)  changes  sign  as  x  passes  through  the 
value  a. 

If  a  change  of  sign  does  take  place  we  have,  in  accordance 
with  Art.  97,  a  maximum  if,  when  x  passes  through  the  value 
a,  the  change  of  sign  is  from  -h  to  —  ;  that  is,  if  fix)  is  a  de- 
creasing function,  and  a  minimum  if  the  change  of  sign  is  from 
—  to  +,  in  which  case  f'{x)  is  an  increasing  function. 

103.  In  many  cases  we  are  able  to  distinguish  maxima  from 
minima  by  examining  the  expression  for  f'{x),  as  in  the  fol- 
lowing examples. 


104  MAXIMA   AND   MINIMA.  [Art.  IO3. 

Given  /W  =  l4^' 

whence  /'(^)  =  l^|£_-_i  ,. 

f\x)  —  O      gives      log  X  —  \y      or      ;ir  =  f. 

Since  log;r  is  an  increasing  function,  it  is  obvious  that,  as  x  in- 
creases through  the  value  ^yf'{x)  increases  ;  it  therefore  changes 
sign  from  —  to  +,  and  consequently  f{e)  is  a  minimum  value 
of/W. 

104.  If  f\x)  does  not  change  sign  we  have  neither  a  maxi- 
mum nor  a  minimum  ;  thus,  let 

f{x)  =  ;i;  —  sin;i:, 
whence  f'{^)  =  i  —  cosjir. 

In  this  case  /'{x)  becomes  zero  when  x  =  2nn^  n  being  zero 
or  any  integer,  but  does  not  change  sign,  since  i  —  cos;t-  can 
never  be  negative ;  consequently  fix)  has  neither  maxima 
nor  minima  values,  but  is  an  increasing  function  for  all  values 
of  X. 

Alternate  Maxima  and  Minima, 

105.  Let  the  curve 

be  constructed,  and  suppose  it  to  take  the  form  represented  in 
Fig.  12.     There  is  a  maximum  value  of 
f(x)   at  B,  another  at  D,   and    minima 
values  occur  at  ^,  at  C,  and  at  E. 

It  is  obvious  that  in  a  continuous  por- 
tion of  the  curve  maxima  and  minima 
ordinates   must   occur    alternately,   and  ^ 

must  separate  the  curve  into  segments 
in  which  the  ordinate  is  alternately  an 
increasing  and  a  decreasing  function  ;  hence,  if  f(x)  has  maxi- 


^ X 


§  XVI.]  ALTERNATE   MAXIMA   AND  MINIMA.  IO5 

ma  and  minima  values,  they  must  occur  alternately  unless  infi- 
nite values  of  the  function  iiitervene.  It  is  also  evident,  with 
the  same  restriction,  that  a  maximum  is  greater  in  value  than 
either  of  the  adjacent  minima,  but  not  necessarily  greater  than 
any  other  minimum  ;  thus,  in  Fig.  12,  the  maximum  at  B  is 
greater  than  the  minima  at  A  and  C,  but  not  greater  than 
that  at  E. 

106.  As  an  illustration  let  us  take  the  following  function  in 
which  it  is  easy  to  discriminate  between  the  maxima  and  min- 
ima values. 

f{x)  =  x{x^df(x-ay. 
Whence, 

f\x)^{x  +  aj  {x  -  ay  +  2x{x  ^  a)(x  -  af  +  ix{x  +  a)'  {x  -  a)\ 

=={x-\-a){x-  ay  {6x'  +  ax  -  a'). 

a  and  —  a  are  evidently  roots  of  f\x)  =  o ;  the  roots  derived 
by  putting  the  last  factor  equal  to  zero  and  solving  are  —  ^a 
and  ^a.     Hence  /'(x)  can  be  written  in  the  form 

f'{x)  =  6{x  +  a)  {x  +  ia)  {x  -  ia)  {x  -  d)\ 

in  which  the  factors  are  so  arranged  that  the  corresponding 
roots  are  in  order  of  magnitude. 

When  X  <  —  a,  f'{x)  is  negative,  and,  if  we  regard  x  as  in- 
creasing continuously,  f\x)  changes  sign  when  x  =  —  a,  when 
X  =  —  ^a,  and  again  when  x  =  ^a,  but  not  when  x  =  a. 

Since  /'{x)  is  at  first  negative  it  changes  sign  from  —  to  + 
when  it  first  passes  through  zero,  that  is  when  x  =  ~  a;  the 
corresponding  value  of  /(x)  is  therefore  a  minimum.  Accord- 
ingly the  value  of  /(x)  corresponding  to  the  next  root  x  =  —  ia 
is  a  maximum,  and  that  corresponding  to  x  =  ^a  is  another 
minimum ;  but  there  is  neither  a  maximum  nor  a  minimum 
corresponding  to  x  :=  a. 


I06  MAXIMA   AND  MINIMA,  [Art.   lO/. 

107.  When  the  function  is  continuous  as  in  the  above  ex- 
ample, that  is,  does  not  become  infinite  for  any  finite  value  of 
Xy  it  is  always  easy  to  determine  by  examining  the  function 
itself  whether  the  last,  or  greatest  value  of  x  in  question,  gives 
a  maximum  or  a  minimum.  Thus,  in  the  above  example,  f{x) 
evidently  increases  without  limit  as  x  increases  without  limit ; 
therefore,  the  last  value  must  be  a  minimum. 


K 


The  Employment  of  a  Substituted  Function, 


108.  Since  an  increasing  function  of  a  variable  increases  and 
decreases  with  the  variable,  such  a  function  will  pass  from  a 
state  of  increase  to  a  state  of  decrease,  or  the  reverse,  simulta- 
neously with  the  variable ;  that  is,  it  will  reach  a  maximum  or 
a  minimum  value  at  the  same  time  with  the  variable. 

This  fact  often  enables  us  to  simplify  the  determination  of 
maxima  and  minima  by  substituting  an  increasing  function  of 
the  given  function  for  the  given  function  itself.  For  example, 
if  we  have 

f{x)  =  V{b'  -f  ax)  +  V{b^  -  ax\ 

we  may  with  advantage  employ  the  square  of  the  given  func- 
tion.    The  square  is 

2b''  ^2s/{b'-a'x% 

which  is  obviously  a  maximum  when  x  —  o^  and,  since  the  square 
of  a  positive  quantity  is  an  increasing  function,  we  infer  that 
f(x)  is  likewise  a  maximum  for  the  same  value  of  x, 

109.  A  decreasing  function  of  the  given  function  may  also 
be  employed  ;  but,  in  this  case,  since  the  substituted  function 
decreases  with  the  increase  of  the  given  function  and  increases 
v/ith  its  decrease,  a  maximum  of  the  substituted  function  indi- 


§  XVI.]  SUBSTITUTED  FUNCTIONS.  lO/ 

cates  a  minimum,  and  a  minimum  indicates  a  maximum  of  the 
given  function. 

Thus,  if  we  have 

X 


f{x)  = 


x'  —  3x  -\-  i' 


the  reciprocal  may  be  employed.  The  reciprocal  of  this  func- 
tion is 

x^  —  2,x  -{-  I  I 

X  X 

whence,  taking  the  derivative,  we  obtain 

^  _  ;ir2  —  I 

*^    :?      ~x~ ' 

which  vanishes  when  x  =  ±  i. 

Since  x^  is  an  increasing  function  when  x  is  positive,  this  deriv- 
ative is  evidently  an  increasing  function  when  x  =  i.  The  re- 
ciprocal is  therefore  a  minimum  for  this  value  of  x,  and  conse- 
quently f(\)  is  a  maximum  value  of  fix).  In  a  similar  manner 
it  may  be  shown  that  /(—  i)  is  a  minimum. 

Examples    XVI. 

Determine  the  maxima  and  minima  of  the  following  functions  : 
I-  f  \x)  =  ^.  .A  min.  for  jt  =  - . 

2.  f\x)  =  — ^  .  A  max.  for  x  =  f^. 


3.  /(x)  —  — _3-^^  .  A  min.  for  x  =  ^a. 

/4./W^  /^^\^^^y*  Amin.  for^=  -  ^V- 


^OS  MAXIMA    AND  MINIMA.  [Ex.  XVI. 

*/  5-  /W  =  sin  2x  —x.  A  max.  for  x  =  nTr  +  ^tt  ; 

a  min.  for  x  =  nTt  ~  ^jt. 

^  6.  /(^)  =  2:v'  +  3x'  —  2>6x  +  12.  A  max.  for  .^  =  -  3  ; 

a  min.   for  x  ^=  2. 

J  7.  f{x)  =  x'  —  sx'  —  gx  +  S'  A  max.  for  jt;  =  —  i  ; 

a  min.  for  x  =  z. 
J 

8.  f{x)  =  3^'— i25.jt'  +  2i6ojc.      A  max.  for  ;»:=:— 4  and  x=:^  ; 

a  min.  for  ^=—3  and  .^=4, 

v   g.  f(x)  ^=  b  +  c{x  —  a)^.  Neither  a  max.  nor  a  min. 

yl  10.  /(^)  =  (^  —  i)*  (jc  +  2)'.  A  max.  for  ;<;  =  ~~  t  5 

a  min.  for  x  =  i. 

y  II.  /(x)  =  {x  —  gY  {x  —  8)*.  A  max.  for  jc  =  8  ; 

a  min.  for  x  =  -8f . 

/  - .    ^         I  —  :r  +  .;«?' 

^    12.    fix 


f  {oc)  =  — ; ^ .  A  min.  for  x  =  h 

•^  ^    ^        1  +  X  —  x^  ^ 

/^  /  N  ax  c^      A  ^  Max.  for  ;<:  =  i ; 

j  Min.  f or  ^  =  ~  i  (^  being  positive). 

V     I5./(^)  =  (I+^I)(7_^)^ 

iSi?/z'(?  by  putting  x  =^  z^.     For  method  of  discriminating  between  max- 
ima and  minima,  see  Art.  107.  Min.  for  ^  =  o,  and  ^  =  7  ; 

Vmax.  for  .x=:  i. 
16.  f{x)  —  5^1?'  +  12^'  —  15^  —  4o:r'  +  \^x^  +  60^  +  27. 

Min.  for  ^  =  —  2. 

J       17.  f{pc)  =  jc"  —  6^*  4-  4^^  +  9^'  —  ^2x  +  3. 

Min.  for  :r  =  —  2,  and  .^  =  i; 
max.  for  .%:  =  —  i . 

^Q/    t  •  • 


§  XVI.]  EXAMPLES.  109 

y  18.  The  top  of  a  pedestal  which  sustains  a  statue  a  feet  in  height  is 
b  feet  above  the  level  of  a  man's  eyes  ;  find  his  horizontal  distance  from 
the  pedestal  when  the  statue  subtends  the  greatest  angle. 

When  the  distance  =  \/\b[a  +  h)\. 

19.  It  is  required  to  construct  from  two  circular  iron  plates  of  radius 
a  a  buoy,  composed  of  two  equal  cones  having  a  common  base,  which 
shall  have  the  greatest  possible  volume. 

The  radius  of  the  base  =  \a  4/6. 

i/  20.  The  lower  corner  of  a  leaf  of  a  book  is  folded  over  so  as  just  to 
reach  the  inner  edge  of  the  page  ;  find  when  the  crease  thus  formed  is 
a  minimum. 
Solution : — 

Let  J/  denote  the  length  of  the  crease,  x  the  distance  of  the  corner 
from  the  intersection  of  the  crease  with  the  lower  edge,  and  a  the 
width  of  the  page. 

By  means  of  the  relations  of  similar  right  triangles,  the  following 
expression  is  deduced  : 

_        X  Vx 
^~  V{x-lay 
Whence  we  obtain 

x=%a, 

which  gives  a  minimum  value  oi  y. 
yy        ?i.  Find  when  the  area  of  the  part  folded  over  is  a  minimum. 

When  X  ='^a. 


XVII. 

The  Employment  of  Derivatives  Higher  than  the  First. 

MO.  To  ascertain  whether /'(;tr)  is  an  increasing  or  a  de- 
creasing function,  (and  thence  whether /(;r)  is  a  minimum  or  a 
maximum),  it  is  frequently  necessary  to  find  the  expression  for 
its  derivative, /"(;ir).  Now,  \i  f'\d)  is  found  to  have  a  positive 
value,  it  follows  that  f\x)  is  an  increasing  function  when  x  —  cu 


no  MAXIMA   AND  MINIMA.  [Art.  I  lO. 

and,  as  was  shown  in  Art.  102,  that/(rt)  is  a  minimum.  On  the 
other  hand,  if  we  find  \.h^tf"(a)  has  a  negative  vakie,  it  follows 
that/'(^)  is  a  decreasing  function,  and  that/"(^)  is  a  maximum. 
To  illustrate,  let 

f{x)  —  ix'  —  iGx"  —  Gx"  +  12, 

then  f\x)  —  \2x''  —  ^Zx"  —  \2x. 

The  roots  of  fix)  —  o  are  x  ~  o,  and  x  —  2  ±  VS- 

In  this  case  /"{x)  =  i^x""  —  g6x  —  12, 

hence  /"(o)  =  —  12  ; 

/{x)  is  therefore  a  maximum  when  ;ir  =  o. 

It  is  unnecessary  to  find  the  values  of  y"(;ir)  for  the  other 
roots ;  for,  since  the  function  does  not  admit  of  infinite  values, 
the  maxima  and  minima  occur  alternately.  The  root  2  —  V  S 
being  negative  and  2  +  4/5  positive,  the  root  zero  is  intermediate 
in  value,  and  therefore  both  the  remaining  roots  give  minima. 

[|(,  If  /'(x)  contains  a  positive  factor  which  cannot  change 
sign,  this  factor  may  be  omitted  ;  since  we  can  determine 
whether  /'{x)  increases  or  decreases  through  zero  by  examin- 
ing the  sign  of  the  derivative  of  the  remaining  factor.    Thus,  if 

Since  z- ^r^  is  always  positive,  we  have  only  to  determine 

(i  4-  ^  ) 

whether  the  factor  i  —  x"^  changes  sign.     Denoting  this  factor 

by  V,  and  putting  v  =  o,  wq  have 

x=  ±1. 

Now  -r-  =  —  2X 

ax 
which  is  negative  for  ;if  =  i  and  positive  for  x——\.     These 


§  XVII.]  EMPLOYMENT  OF  SECOND   DERIVATIVES.  Ill 

roots,  therefore,  give  respectively  a  maximum  and  a  minimum 
value  o{  f{x). 

(12.  There  may  be  roots  of  the  equation  f'{x)  —  o  which 
correspond  to  neither  maxima  nor  minima,  since  it  is  a  condi- 
tion essential  to  the  existence  of  such  values  that  f\x)  shall 
change  sign.  When  such  cases  arise,  the  form  assumed  by  the 
curve  y  —  f(x)  in  the  immediate  vicinity  of  the  point  at  which 
X  ^=^  a  will  be  one  of  those  represented 
at  A  and  B  in  Fig.  13- 

At  these  points  the  value  of  tan  ^  or 
f\x)  is  zero,  but  at  A   it   is  positive  on 
both  sides  of  the  point,  and  fix)  or  y  is 
an  increasing  function,  while  at  B  fix)  0 
is  negative  on  both  sides  of  the  point,  Fig.  13. 

and  f{x)  is  a  decreasing  function. 

il3.  It  is  important  to  notice  that  at  A  the  value  zero 
assumed  by  f\x)  constitutes  a  minimum  value  of  this  function, 
thus  a  root  of  f'{x)  —  o  for  which  /'{^)  is  a  niinimuni  corre- 
sponds to  a  case  in  which  f{x)  is  an  increasing  function.  In 
like  manner  a  root  of  f\x)  =  o  for  which  /'{x)  is  a  maximum 
is  a  case  in  which  f{x)  is  a  decreasing  function. 

H4-.  It  follows  from  the  preceding  article  and  from  Art. 
102  that,  if/'(^)  =  O,  then,  of  the  two  functions /(;ir)  and/'(.r), 
one  will  be  a  maximum  and  the  other  a  decreasing  function, 
or  else  one  will  be  a  minimiun  and  the  other  an  increasing 
function.  Hence,  if  we  consider  the  case  in  which  the  given 
function  and  several  of  its  successive  derivatives  vanish  for  the 
same  value  of  x,  it  is  evident  that  when  these  functions  are 
arranged  in  order  they  will  be  either  alternately  maxima  and 
decreasing  functions^  or  alternately  minima  and  increasing  func- 
tions. 

lis.   Now  suppose  that  ^{x)  is  the  first  of  these  successive 


112  MAXIMA    AND  MINIMA.  [Art.   1 1 5. 

derivatives  that  does  not  vanish  when  x  =  a^  then,  writing  the 
series  of  functions 

/W,  /'W.  /"W. /'-"W,  /V), 

let  us  assume  first  that  f"{a)  is  positive.  Then  in  the  above 
series  of  functions  f"~\ci),  f~\<^\  ^tc,  will  be  increasing 
functions  while  /"~X<^),  f"~\a),  etc.,  will  be  minima. 

Now  whenever  7t  is  odd,  the  original  function  will  belong  to 
the  first  of  these  classes  and  will  be  an  increasing  function, 
while  if  n  is  even  the  original  function  will  belong  to  the  second 
class  and  will  be  a  minimum. 

On  the  other  hand,  if  f'\d)  has  a  negative  value,  the  series 
of  functions  will  be  alternately  decreasing  functions  and  maxi- 
ma ;  and  when  n  is  odd  f{a)  will  be  a  decreasing  function,  but 
when  n  is  even  f{d)  will  be  a  maximum. 

Thus  we  shall  have  neither  maxima  nor  minima  unless  the 
first  derivative,  which  does  not  vanish  when  x  —  a,  is  of  an 
even  order ;  but  when  this  is  the  case  we  shall  have  a  maximum 
or  a  minimum  according  as  the  value  of  this  derivative  is  nega- 
tive or  positive. 

116.  The  following  function  presents  a  case  in  which  the 
above  principle  is  advantageously  employed. 

f{x)  =  £-^  +  £~"^   +  2  cos  X, 

f'{x)  =  £^  —  e~^  —  2  sin  x. 

Zero  is  evidently  a  root  of  the  equation  /'{x)  —  o.*  In  this 
case 

*  Zero  is  the  only  root  of  /'{x)  =  o  in  this  example  ;  for 
/  (x)  = > ' . 

f"{x)  therefore  cannot  be  negative,  hence  f'{x)  cannot  again  assume  the  value 
zero. 


§  XVII.]     INFINITE    VALUES  OF   THE  DERIVATIVE.  II3 

f"{x)  =  f"^  +  f""^  —  2  cos;ir     .'.     /"(o)  =  O, 

f"\x)  =  €^  —  a~^  4-  2  sin  ;ir     .-.     /'"(o)  =  O, 

/''(x)  =  6^  -{■  €~^  +  2  COS  ;ir     .-.      /'^  (o)  =  4. 

The  fourth  derivative  being  the  first  that  does  not  vanish,  and 
having  a  positive  value,  we  conclude  that  x  =  o  gives  a  mini- 
mum value  of  /{x). 


Infinite  Values  of  the  Derivative, 

(17.  It  was  shown  in  Art.  98  that  if  we  have,  for  x  =  a^ 

f\x)  =   «, 

a  maximum  value  will  present  itself  \i  f\x)  changes  sign  from 
+  to  — ,  and  a  minimum  if  it  changes  sign  from  —  to  +.  It 
may,  however,  happen  in  these  cases  that  the  value  of  f  (a)  is 
also  infinite.     When/(^)  is  finite,  the  form  of  the  curve 

in  the  vicinity  of  a  maximum  or  a  minimum   ordinate   of  this 
variety  is  represented  at  A  and  B  in  Fig.  14. 
As  an  example,  let 


whence 


/(;r)=(;r*  -  /^i)* 
f\x)  =  %x'\x^-lyf)'^. 


k 


fix)  is  infinite  when  x  =0  and  when  x  =  b. 
When    X  =  o    fix)    does    not    change     sign, 
since  x~^  cannot  be  negative,  but  when  x  =  b  q 
it  changes  sign  from  —  to  +  ;  hence  fix)  has 
a  minimum  value  when  x  =.  b. 


X 


Fig.  14. 


\/ 


114  MAXIMA   AND  MINIMA.  [Ex.  XVII. 

Examples    XVII. 

y         I.  Show  that  ae^'  +  bs~^'  has  a  minimum  value  equal  to  2  V{(ib). 

Find  the  maxima  and  minima  of  the  following  functions  : 

2.  f(x)  =  X  sin  X. 

A  maximum  for  a  value  of  x  in  the  second  quadr^int  satisfying  the 
equation  tan  x  ^=  —  x. 

J  ..   .       a^  b^ 


3-  /(^)  =  -  +  - 


X      a 


The  roots  x  = r  and  x  = r  give  a  min.  and  a  max.  if  b  is 

«  +  /^  a  —  b^ 

positive,  but  a  max.  and  min.  if  b  is  negative. 

^     4.  /(jc)  =  2  cos  X  +  sin'^  .r. 

Solution  : —  f  {.^^  =^  2  sin  jjc  (cos^  —  i)  J 

rejecting  the  factor  2(1  —  cos  x\  which  is  always  positive,  we  put 
V  —  —  sm  X.         Hence  -^  =  —  cos  x. 


y 


A  max.  for  .;c  =  2n7t ; 

a  min.  for  x  =  (2;?  +  i)  rr. 

5.  /(jc)  =  sin  x{i  -\-  cos  ^).  A  max.  for  ^  =  Jtt  ; 

a  min.  for  x=^  —  \7t ; 
neither  for  x  =  7t, 


j  6.  f  {x)  =  sec  X  +  log  cos''  x. 

Multiplying  the  derivative  by  cos^  x^  we  obtain 

,^  .   z;  =  sinj[:(i  —  2  cos^i;). 

c^  A  A  max.  for  ^  =  o,  and  x-=-  n  \ 

n-ftw  ,     ^  -H^  <  a  min.  for  ^  =  ±  J  tt. 

xf  \  __   tan'  ^  A  min.  for  ^  =  o,  |7r,  |;r,  and  n  ; 

'  ""^  ^  tan  3JIC  *  a  max.  for  Jt:  =  ^tt,  \7t^  ^tt,  etc. 

J   8.  /(x)  =  £"  +  f-*  —  .rl  A  min.  for  x  =  o. 


§  XVII.]  EXAMPLES.  115 

/ : 

9.  Find  maxima  and  minima  of  the  following  functions  : 

f  {pc)  =  {x^  —  I^)  ^.  A  min.  for  ^  =  o. 


/ 


10.  f{pc)  =  {x-  —  l>^)-^.  A  max.  iox  x  =  o  \ 

a  min.  for  x  =  ^l  l^* 


II.  f{x)  —  {pc^  +  3^  +  2)^  +  ^\ 

f\x)  =  00  gives  min.  corresponding  to  ^=  —  2,  x—  —  i  and  jc=o. 
f'{x)  =  o  gives  two  intermediate  maxima. 

^12.  /{x)  =  {x^  +  2^)^  —  {x  +  3)^.     Max.  for  x=i{—^±  4/17)  ; 

min.  for  x  ^=  o  and  a=  —  2. 

I  /    13.  /(:*:)  =  (^  —  ^)5  (^x  —  ^)^  +  ^.  A  max.  for  x  — ; 

min.  for  x  =^  a  and  x  ^=^  b. 


/  14.  /(^)  =  {x-a){x-b) 


A  mm.  for  x  — 

a  +  I 


IS.  f{x)  =  {x-  a)i  (x  -  b)\ 

Solutions  iox  X  =  a  and  x  =  l{2b  +  a)  ;  if  b  >  a,  the  former  gives 
a  max.  and  the  latter  a  min. 


t 


Miscellaneous  Examples. 


Max.  for  ^  =  4. 

J^/   X       -^'^  —  Jc  +  I  A  max.  for  ^  =  o  ; 

'       x^  +  X  —  I  a  mm.   for  .v  —  2. 


Il6  MAXIMA   AND   MINIMA.  [Ex.  XVII. 

7 ~~ 

3-  f{^)  =  ■^'""  ^**-  A  min.  for  jc  =  ^  . 

/    4.  The  equation  of  the  path  of  a  projectile  being 
y  =  X  tan  a 


4^  cos"^<^ ' 


find  the  value  of  x  when  j^^  is  a  maximum  ;  also  the  maximum  value 
of  y.  Max.  when  x  ■=^  h'ivci2a^  and  y  =  h  sin^  a. 

\  5.  In  a  given  sphere  inscribe  the  greatest  rectangular  parallel- 
epiped. 

Solution : — 

Regarding  any  one  edge  as  of  fixed  length,  it  is  easy  to  show  that 
the  other  two  edges  are  equal.     Hence  the  three  edges  are  equal. 

^  6.  In  a  given  cone  inscribe  the  greatest  rectangular  parallelo- 
piped. 

Solution  : — 

Regarding  the  parallelopiped  as  inscribed  in  a  cylinder  which  is 
itself  inscribed  in  the  cone,  the  base  is  evidently  a  square,  and  the 
altitude  is  that  of  the  maximum  cylinder.     See  Ex.  XV,  9. 

V  7.  A  Norman  window  consists  of  a  rectangle  surmounted  by  a 
semicircle.  Given  the  perimeter,  required  the  height  and  breadth  of 
the  window  when  the  quantity  of  light  admitted  is  a  maximum. 

The  radius  of  the  semicircle  is  equal  to  the  height  of  the  rectangle. 

>J  8.  A  tinsmith  was  ordered  to  make  an  open  cylindrical  vessel  of 
given  volume,  which  should  be  as  hght  as  possible  ;  find  the  ratio  be- 
tween the  height  and  the  radius  of  the  base. 

The  height  equals  the  radius  of  the  base. 

^  9.  What  should  be  the  ratio  between  the  diameter  of  the  base  and 
the  height  of  cylindrical  fruit-cans  in  order  that  the  amount  of  tin  used 
in  constructing  them  may  be  the  least  possible  ? 

The  height  should  equal  the  diameter  of  the  base. 


§  XVII.]  MISCELLANEOUS  EXAMPLES.  11/ 

V  lo.  Determine  the  circle  having  its  centre  on  the  circumference  of 
a  given  circle  so  that  the  arc  included  in  the  given  circle  shall  be  a 
maximum. 

A  max.  for  the  value  of  0  which  is  in  the  first  quadrant. 


/„ 


/x 


Given  the  vertical  angle  of  a  triangle  and  its  area  ;  find  when 
its  base  is  a  minimum.  The  triangle  is  isosceles. 

\/i2.  Prove  that,  of  all  circular  sectors  of  the  same  perimeter,  the 
sector  of  greatest  area  is  that  in  which  the  circular  arc  is  double  the 
radius. 


13.  Find  the  minimum  isosceles  triangle  circumscribed  about  a  par- 
abolic segment. 

The  altitude  of  the  triangle  is  four-thirds  the  altitude  of  the  seg- 
ment. 


1/14. 


Find  the  least  isosceles  triangle  that  can  be  described  about  a 
given  ellipse,  having  its  base  parallel  to  the  major  axis. 

The  height  is  three  times  the  minor  semi-axis. 

15.  Inscribe  the  greatest  parabolic  segment  in  a  given  isosceles 
triangle. 

The  altitude  of  the  segment  is  three-fourths  that  of  the  triangle. 

16.  A  steamer  whose  speed  is  8  knots  per  hour  and  course  due  north 
sights  another  steamer  directly  ahead,  whose  speed  is  10  knots,  and 
whose  course  is  due  west.  What  must  be  the  course  of  the  first  steamer 
to  cross  the  track  of  the  second  at  the  least  possible  distance  from  her  ? 

N.  53°  8'  W, 

17.  Determine  the  angle  which  a  rudder  makes  with  the  keel  of  a 
ship  when  its  turning  effect  is  the  greatest  possible. 

Solution : — 

Let  ^  denote  the  angle  between  the  rudder  and  the  prolongation 
of  the  keel  of  the  ship  ;  then  if  b  is  the  area  of  the  rudder  that  of  the 
stream  of  water  intercepted  will  be  /^  sin  ^  :  the  resulting  force  being 
decomposed,  the  component  perpendicular  to  the  rudder  contains  the 
factor  sin''  ^.  Again  decomposing  this  force,  and  taking  the  compo- 
nent that  is  perpendicular  to  the  keel  of  the  ship,  which  is  the  only 


Il8  MAXIMA    AND  MINIMA.  [Ex.  XVII. 

part  of  the  original  force  that  is  effective  in  turnii.g  cAc  ship,  the  ex- 
pression to  be  made  a  maximum  is 

sin"  ^  cos  ^. 
Whence  we  obtahi 

tan  ^  —  |/2. 

1 8.  The  work  of  driving  a  steamer  through  the  water  being  propor- 
tional to  the  cube  of  her  speed,  find  her  most  economical  rate  per  hour 
against  a  current  running  a  knots  per  hour. 

Solution  : — 

Let  z;  denote  the  speed  of  the  steamer  in  knots  per  hour.  The 
work  per  hour  will  then  be  denoted  by  kv^,  k  being  a  constant,  and  the 
actual  distance  the  steamer  advances  per  hour  hy  v  ^  a.  The  work 
per  knot  made  good  is  therefore  expressed  by 


Whence  we  obtain  the  result 


X^i^  h^ 


CHAPTER   VII. 

The  Development  of  Functions  in  Series. 


XVIII. 

The  Nature  of  an  Infinite  Series, 

1(8.  A  FUNCTION  which  can  be  expressed  by  means  of  a 
limited  number  of  integral  terms,  involving  powers  of  the  inde- 
pendent variable  with  positive  integral  exponents  only,  is  called 
a  rational  integral  function. 

When  f(x)  is  not  a  rational  integral  function,  it  is  usually 
possible  to  derive  an  unlimited  series  of  terms  rational  and  in- 
tegral with  respect  to  x,  which  may  be  regarded  as  an  algebraic 
equivalent  for  the  function.  The  process  of  deriving  this  series 
is  called  the  development  of  the  function  into  an  infinite  series. 

When  the  given  function  is  in  the  form  of  a  rational  frac- 
tion, the  ordinary  process  of  division  (the  dividend  and  divisor 
being  arranged  according  to  ascending  powers  of  x)  suffices  to 
effect  the  development.     Thus — 

-i-i-^  =  I  4-  2;ir  +  2;ir'  +  2;tr'  +    •    •   •   , 


a  series  of  terms  arranged  according  to  ascending  powers  of  Xy 
each  coefficient  after  the  absolute  term  being  2. 

It  is  to  be  observed,  in  the  first  place,  that,  owing  to  the 
indefinite  number  of  terms  in  the  second  member,  the  equa- 
tion as  written  above  cannot  be  verified  numerically  for  an 
assumed  value  of  x»     In  this  case,  however,  the  process  not 


I20  THE  DEVELOPMENT  OF  FUNCTIONS       [Art.    1 1 8. 

only  gives  us  the  series,  but  the  remainder  after  any  number  of 
terms.  Thus  carrying  the  quotient  to  the  term  containing  x^, 
and  writing  the  remainder,  we  have 

—  \  -^  2X  -^  2X'^    '    '    '     -^  2X''  -\ . 


1  --  X 

This  equation  may  now  be  verified  numerically  for  any  assumed 
value  of  X ;  or  algebraically  by  multiplying  each  member  by 
1  —  X,  thus  obtaining  an  identity. 

The  ordinary  process  of  extracting  the  square  root  of  a 
polynomial  furnishes  an  example  of  a  series  which  may  be  ex- 
tended so  as  to  include  as  many  terms  as  we  please  ;  but  this 
process  gives  us  no  expression  for  the  remainder. 

119.  Assuming  that  /(x)  admits  of  development  into  a 
series  involving  ascending  powers  of  x,  and  denoting  the  re- 
mainder after  7i  -\-  i  terms  by  R,  we  may  write 

f{x)  =A  +  Bx+  Cx'+  .  .  .    +  Nx^'  +  i?, .     .     .     (i) 

in  which  A,  By  C,  ...  N  denote  coefficients  independent  of  Xy 
and  as  yet  unknown  ;  the  value  of  R  is  however  not  indepen- 
dent of  X.  If  the  coefficients  By  C,  .  .  .  N  admit  of  finite 
values,  it  may  be  assumed  that  i?  is  a  function  of  x  which  van- 
ishes when  X  =  O'y  and  in  accordance  wifh  this  assumption 
equation  (i)  becomes,  when  x  —  o, 

/iO)     =     Ay (2) 

which  determines  the  first  term  of  the  series.  If  in  any  case 
the  value  of  /(o)  is  found  to  be  infinite,  we  infer  that  the  pro- 
posed development  is  impossible. 

120.  When  the  coefficients  B,  C,  ,  .  .  N  admit  of  finite 
values,  and  the  value  of  the  function  to  be  developed  remains 


%  XVIII.]  XN FINITE   SERIES.  C^2I 

finite,  R  will  have  a  finite  value.  If  moreover  the  value  of  R 
decreases  as  n  increases,  and  can  be  made  as  small  as  we  please, 
by  sufficiently  increasing  n^  the  series  is  said  to  be  convergent, 
and  may  be  employed  in  finding  an  approximate  value  of  the 
function  f{x)  ;  the  closeness  of  the  approximation  increasing 
with  the  number  of  terms  used.  A  series  in  which  R  does  not 
decrease  as  n  increases  is  said  to  be  divergent. 

When  the  successive  terms  of  a  series  decrease  it  does  not 
necessarily  follow  that  the  series  is  convergent ;  for  the  value 
of  the  equivalent  function,  and  consequently  that  of  R^  may  be 
infinite.     To  illustrate,  if  we  put  x  —  \  m  the  series 

X  +  J^'^  ^\x'  ^\x'  ^'   .   .  , 
we  obtain  the  numerical  series 

it  can  be  shown  that,  by  taking  a  sufficient  number  of  terms,  the 
sum  of  this  series  may  be  made  to  exceed  any  finite  limit,  the 
value  of  the  equivalent  or  generating  function  of  the  above  series 
being  in  fact  infinite  when  x  =  i.* 

121.  Since  R  vanishes  with  x,  every  series  for  which  finite 
coefficients  can  be  determined  is  convergent  for  certain  small 
values  of  x.  In  som-e  cases  there  are  limiting  values  of  x,  both 
positive  and  negative,  within  which  the  series  is  convergent, 
while  for  values  of  x  without  these  limits  the  series  is  diver- 
gent.    These  values  of  x  are  called  the  limits  of  convergence. 

*  If  we  consider  the  first  two  terms  separately,  and  regard  the  other  terms  as 
arranged  in  groups  of  two,  four,  eight,  sixteen,  etc.,  the  groups  will  end  with  the 
terms  \,  \,  -j^g,  3^,  etc.  The  sum  of  the  fractions  in  the  first  group  exceeds  f  or  ^, 
the  sum  of  those  in  the  second  exceeds  \  or  \,  and  so  on  ;  hence  the  sum  of  2iV  such 
groups  exceeds  the  number  N,  and  N  may  be  taken  as  large  as  we  choose. 

The  generating  function  in  this  case  is  log ,  and  unity  is  the  limit  of  con- 
vergence. 


122  THE  DEVELOPMENT  OF  FUNCTIONS.      [Art.    121. 

We  shall  now  demonstrate  a  theorem  by  which  a  function 
in  the  form  f{xo  +  h)  may  be  developed  into  a  series  involving 
powers  of  h^  and  in  Section  XIX  we  shall  show  how  this 
theorem  is  transformed  so  as  to  give  the  expansion  of  f{x)  in 
powers  of  x. 


Taylor^s   Theorem. 

(22.  A  function  of  h  of  the  form  f{xo  -\-  Jt)  in  general  admits 
of  development  in  a  series  involving  ascending  powers  of  h. 
We  therefore  assume 

/(;ro  +  >^)  =  ^o  4-  BJt  +  a>^'+  •   .   .  +  NoU"  +  7?o,  .    .  (i) 

in  which  Aoy  Boj  Co,  .  .  .  iVo  are  independent  of  /i,  while  Ro 
is  a  function  of  /i  which  vanishes  when  /i  is  zero.  Hence,  mak- 
ing ^  =  o,  we  have 

/(^o)  =  Ao. 

We  have  now  to  find  the  values  oi  Bo,  Co,  -  -  -  No,  which 
are  evidently  functions  of  Xo.     For  this  purpose  we  put 

^i  =  Xo-\-  hj         whence         h  ^^  x^—  Xo  ; 

substituting,  equation  (i)  takes  the  form 

f{x;)^f{Xo)^Bo{x,-Xo)^Co{x,-Xoy    '    •    •    -{-JVoix.-Xof-hRoy 

in  which  we  may  regard  Xj,  as  constant  and  Xo  as  variable.  Re- 
placing the  latter  by  x^  and  its  functions,  Bo,  Cc,  .  .  »  iVo,  and 
Rn,  by  B,  C,  .  .  .  Ny  and  R,  we  have 

f{x:)=f{x)-^B{x.^x)  +  C{x.-'xy-'^^N{x.^xT^R,   .   (2) 

Taking  derivatives  with  respect  to  x,  we  have 


§  XVIII.]  TAYLOR'S   THEOREM.  1 23 

-  nN{x.  -  x)-^  +  {X,  -  xy^  +  g  .     .     .     .     (3) 

To  render  the  development  possible,  B,  Cj  .  .  .  JV,  and  R  must 
have  such  values  as  will  make  equation  (3)  identical,  that  is,  true 
for  all  values  of  x, 

123  It  is  evident  that  B  may  be  so  taken  as  to  cause  the 
first  two  terms  of  equation  (3)  to  vanish,  and  that,  this  being 
done,  C  can  be  so  determined  as  to  cause  the  coefficient  of 
(>!  —  x)  to  vanish,  D  so  as  to  make  the  coefficient  of  {x^  —  xj 
vanish,  and  so  on.     The  requisite  conditions  are 

f{x)-B=o,        g-2C=o,        g-3Z)  =  o,etc., 

A  a      u  I  .ndN      dR 

and  finally  ix^  —  x)  — — f-  -7-  =  o. 

'    dx       dx 

From  these  conditions  we  derive 

B=f{x\  C=i^=i/"{x), 

and  in  general  N— /"(x). 

Putting  Xo  for  x,  and  substituting  in  equation  (i)  the  values  of 
Aoy  Boy  Coi  .  .  .  No,  We  obtain 

/(^„  +  A)=/(;r„)+/'(^o)/^+/"(^o)-^  - .  .+/«(;r„)— ^-  +je„.(4) 


124  THE  DEVELOPMENT  OF  FUNCTIONS.      [Art.    1 2 3. 

This  result  is  called  Taylor's  Theorem,  from  the  name  of  its  dis- 
coverer, Dr.  Brook  Taylor,  who  first  published  it  in  171 5. 

It  is  evident  from  equation  (4)  that  the  proposed  expansion  is 
impossible  when  the  given  function  or  any  of  its  derived  func- 
tions is  infinite  for  the  value  x^. 


/ 


Lagrange  s  Expression  for  the  Remainder, 


124.  i?  denotes  a  function  of  x  which  takes  the  value  ^o 
when  X  —  Xof  and  becomes  zero  when  x  —  x^.  It  has  been 
shown  in  the  preceding  article  that  R  must  also  satisfy  the 
equation 

,  .^dN      dR 

or,  substituting  the  value  of  N  determined  above, 

^  =  -  (-^^^^/-(^) (5) 

This  equation  shows  that  —  cannot  become  infinite  for  any 

ax 

value  of  X  between  x^  and  x^^  provided /''"^'(;tr)  remains  finite 
and  real  while  x  varies  between  these  limits.  Since  it  follows 
from  the  theorem  proved  in  Art.  104  that  all  preceding  deriva- 
tives must  be  likewise  finite,  the  above  hypothesis  is  equivalent 
to  the  assumption  that/(;ir)  and  its  successive  derivatives  to  the 
(n  -f  \)th  inclusive  remain  finite  and  real  while  x  varies  from  Xo 
/^  Xc  +  h. 

(25.  Let  P  denote  any  assumed  function  of  x  which,  like 
R^  takes  the  value  Ro  when  x  —  Xo  and  the  value  zero  when 

X  =  x^,  and  whose  derivative  —r-  does  not  become  infinite  or 

dx 

imaginary  for  any  value  of  x  between  these  limits. 


§  XVIIL]      EXPRESSIONS  FOR    THE  REMAINDER.  1 25 

Then,  Ro  being  assumed  to  be  finite^  P  —  R  denotes  a  func- 
tion of  X  which  vanishes  both  when  x  —  Xo  and  when  x  —  Xt. 
and  whose  derivative  cannot  become  infinite  for  any  interme- 
diate value  of  X,  It  follows  therefore  that  the  value  of  this 
function  cannot  become  infinite  for  any  intermediate  value  of  x. 

Since,  as  x  varies  from  x^  to  x^^  P  —  R  starts  from  the  value 
zero  and  returns  to  zero  again,  without  passing  through  infinity, 
its  numerical  value  must  pass  through  a  maximum  ;  hence  its 
derivative  cannot  retain  the  same  sign  throughout,  and  as  it  can- 
not become  infinite  it  must  necessarily  become  zero  for  some 
intermediate  value  of  x.  Since  x-,  =Xo  +  /i  this  intermediate 
value  of  X  can  be  expressed  by  Xo  +  O/i^  0  being  2.  positive  proper 
fraction.  It  is  therefore  evident  that  at  least  one  value  of  x 
of  the  form 

x  =  Xo-^  Qh 

will  satisfy  the  equation 

dP      dR  ,.. 

'dx-^x-"" ^^) 

126.  The  value  of  P  will  fulfil  the  required  conditions  if  we 
assume 


_{Xj—Xf^ 


'R. 


o» 


for  this  function  takes  the  value  ^o  when  x  =  Xo  and  vanishes 
when  X  =^  x^\  moreover  its  derivative  with  reference  to  ;r,  viz., 

dP  _     («  +  iH-r,--i-y' 

Tx-  li^^  ^°'    •   •   •    •   w 

does  not  become  infinite  for  any  intermediate  value  of  x.  Sub- 
stituting in  equation  (6)  the  values  of  the  derivatives  given  in 
equations  (5)  and  (7),  and  solving  for  R^^  we  obtain 

Ro=^ ^^^.~—-,r^\xo^- eh).    ...   (8) 

1.2-  •  •^2.(«  +  l)  ^ 


126  THE  DEVELOPMENT  OF  FUNCTIONS.       [Art.  1 26. 

This  expression  for  the  remainder  was  first  given  by  La- 
grange. 

The  series  may  now  be  written  thus : 

f(x.^  h)  =f{x.)  +f'{x:)h  +f\Xo)  -^  •  ■  • 

It  should  be  noticed  that  the  above  expression  for  the  remain- 
der after  n  +  i  terms  differs  from  the  next,  or  {?i  +  2jth  term 
of  the  series,  simply  by  the  addition  of  6h  to  Xo^ 

The  Binomial  Theorem, 

127.  We  shall  now  apply  Taylor's  Theorem  to  the  function 
(a  +  by  in  order  to  obtain  a  series  involving  ascending  powers 

Qib, 

In  this  case  b  takes  the  place  of  h,  and  a  that  of  Xo  ;  hence 
f{x)^x  /.     f{x^^Xo  .         =a 

\x)  =  mx  .,'.    /\Xo)  =  'mXo  —ma 

f'\x)  =  m{m  —  \)x'"''''    .'.  /"{Xo)=m(m  —  i)x'"~^  =  m{m—  i)a"~' 
and 
/"{xo)  =  m{m  —  i){m  —  2)  -  .  -  (m  -  n  +  i)a'"-''. 

Whence 

(a  +  by"=  a'"+  md'^-'b  -f-  '"^^ILZ^  a!"-'b' 

m{in  -  \){in  -  2)  .  ♦  .  (in  -  n  ■\-  i)^^^..^«  4.  .  ,  . 
1.2.3  '  '  '  f^ 

This  result  is  called  the  Binomial  Theorem. 


J 


§  XVIII.]  EXAMPLES.  127 


^. 


Examples  XVIII. 


To  expand  log  {x^+  h)  by  Taylor's  Theorem. 
Solution : — 

f(x)  -  log  X         :.        /{x^}  =  log  x^ 


/»  =  -^        .-.      /■V„)  =  -i5 


/'»=^  .-.     /"'(^»)  =  -^ 


/"W=-^         •••       /^Vo)=-'-^ 


By  substituting  in  equation  (4),  Art.  123,  we  obtain 

log(^,+  ^)=log^,+  ^-—  +_------...  -(_i)»         4-^0. 

^o  ^^o  Z^o  4-^0  ^^-^o 

Employing  Lagrange's  expression  for  the  remainder   (Art.   126)   we 
derive 


^       2.  Expand  a=^°  +  \ 


{n  +  i)(^o+  ^^T'''' 


Solution : — 


128 


THE  DEVELOPMENT  OF  FUNCTIONS.  [Ex.  XVIII. 


f{x)  =  a' 
fix)  =  log  a-a' 


/'(^o)   ^  log  ^'^"^ 


f\x)  =  (log  aY-a'       .-.       /»(^J  -  (log  ay-a'^ 
Substituting  in  equation  (4),  Art.  123,  we  have 

^..»  ==  ,..  [,  +  log  a-/>  +  (log  aY^...  +  (Mflll 


+   i?.. 


1/      3.  Find  the  expansion  of /(jc^  4-^),  when  /(^)  =  ^  log  x  —  x^  writ' 
ing  the  {n  +  i)'''  term  of  the  series. 


/{x^  +  A)  =  x^  log  x^—x^-h  log  jc„./^  + 


•^o     1-2  -^o  *2-3 


"^  ^       '^<-^  '(n-  i)n 


4.  Expand  sin"'  (x^  +  h)  to  the  fourth  term  inclusive. 


,/           ,\          •       1                    h                      x^  h^ 

sm-'(-^o+  ^)  -  sm-^^,+ -3:  + ^  •  — 

^      I  +  2:r^         /^^      ^ 


(i-OV^-2-3 
y]       5.  Prove  that 

j  I-2-3-4-5  J 

^    6.  Prove  that 

tan  (J;r  +  /^)  =  I  +  2^  +  2/^=  +  P"  +  V"^'  +  •  •  • 


§  XIX.]  MACLAURIN'S  THEOREM.  1 29 

XIX. 

Maclaurin  s   Theorem. 

128.  We  shall  now  give  a  particular  form  of  Taylor's  Series^ 
which  is  usually  more  convenient,  when  numerical  results  are  to 
be  obtained,  than  the  general  form  given  in  the  preceding  sec- 
tion. 

This  form  of  the  series  is  obtained  by  putting  x^=-0  and 
replacing  h  by  x  in  equation  (4),  Art.  123.     Thus, 

/(^)=/(o)+/'(o)^+/"(o)  ^  ■  •  •  +/"(o)  ^    l]^  _  ^  +R^  .  .  (I) 

and,  the  same  substitutions  bding  made  in  equation  (8),  Art.  126, 
we  obtain 


I  -2  •••(«+  l)* 

Equation  (i)  is  called  Maclaurin's  Theorem:  it  maybe  used 
in  developing  any  function  to  which  Taylor's  Theorem  is  ap- 
plicable, by  giving  a  different  signification  to  the  symbol  /. 
Thus,  if  log  (i  +  /t)  is  to  be  developed  by  Taylor's  Theorem, 
/{x)  =  log  Xf  the  value  of  x^  being  unity;  but,  if  log  (i  -\-  x) 
is  to  be  developed  by  Maclaurin's  Theorem,  we  must  put 
/{x)  =  log  (i  +  x),     (Compare  Ex.  XVIII.,  i,  with  Art.  130.) 

The  Exponential  Series  and  the   Value  of  e, 

129.  As  an  example  of  the  application  of  the  above  theo- 
rem, we  shall  deduce  the  development  of  the  function  «-^,  which 
is  called  the  exponential  series,  and  shall  thence  obtain  a  series 
for  computing  the  value  of  f. 

The  successive  derivatives  of  e-^  being  equal  to  the  original 
function,  the  coefficients, y(o),/"(o),  etc.,  each  reduce  to  unity; 


130  THE  DEVELOPMENT  OF  FUNCTIONS.       [Art.   1 29. 

we  therefore  derive,  by  substituting  in  equation  (i)  and   in- 
troducing the  value  of  Ro, 

f^=  I  +;ir+  —  + .  .  •  + +  f^-'' 


2       1-2.3  1-2 n  1-2  ••  .  '{71  +  I) 

Putting  X  equal  to  unity,  we  obtain  the  following  series,  which 
enables  us  to  compute  the  value  of  the  incommensurable  quan- 
tity f  to  any  required  degree  of  accuracy : 

III 

f  =  I  +  I  +  —  +  ■  + •  •  • 

1-2        12-3        I-2-34 


I-2-3  •  •  n      I-2-3  .  •  (;2  +  i)' 

The  computation  may  be  arranged  thus,  each  term  being  de- 
rived from  the  preceding  term  by  division : 

2.5 
,16666666667 
4166666667 

833333333 

138888889 

1984 I 270 

2480159 

275573 

27557 

2505 

209 

16 

I 


2.71828182846 


Since  f*  is  less  than  e,  the  remainder  (n  being  14)  is  less  than 
j\  of  the  last  term  employed  in  the  computation,  and  therefore 
cannot  affect  the  result.  Inasmuch  as  each  term  may  contain  a 
positive  or  negative  error  of  one-half  a  unit  in  the  last  decimal 


§  XIX.]  THE    VALUE    OF  e.  13! 

place,  we  cannot,  in  general,  rely  upon  the  accuracy  of  the  last 
two  places  of  decimals,  in  computations  involving  so  large  a 
number  of  terms.  Accordingly,  this  computation  only  justifies 
us  in  writing 

£=2.718281828. 


Logarithmic  Series. 

130.  The  logarithmic  series  is  deduced  by  applying  Mac- 
laurin's  Theorem  to  the  function  log(i  +  x). 
In  this  case 


/w  = 

:log(l    +   X)    . 

••     /(o)  =  o 

/w  = 

I 

I   +  X 

••    /'(o)=i 

/"W  = 

I 

••    /"(o)  =  -r 

(l+^y          ■ 

/'"W  = 

1-2 

■.    /'"(O)  =  1-2 

f\x)  = 

I-2-3 

.    /-(o)=- 1.2.3, 

X  X  X 

hence  loo^  (i  -\- x)  =  x -H +  .  . .      .     .     .     (i) 

•   ^^  ^  234  ^  ^ 

Since  this  series  is  divergent  for  values  of  x  greater  than 
unity  (see  Art.  120),  we  proceed  to  deduce  a  formula  for  the 
difference  of  two  logarithms,  which  may  be  employed  in  com- 
puting successive  logarithms;  that  is,  denoting  the  numbers 
corresponding  to  two  logarithms  by  n  and  n  +  k,  we  derive  a 
series  for 

log  {n  ■\-  h)  —  log  n  =  log . 


132  THE  DEVELOPMENT  OF  FUXCTIONS.       [Art.  I30. 

A  series  which  could  be  employed  for  this  purpose  might  be 

ft  ~\~  Ii  h 

obtained  from  (i),  by  putting in  the  form  i  +  -.    We  ob- 

n  n 

tain,  however,  a  much  more  rapidly  converging  series  by  the 
process  given  below. 

Substituting  —  ;ir  for  ;ir  in  (i),  we  have 

234 
log  (l  —  ;r)  =  —  ;ir ^  _  .  .  .     .     .     (2) 


Subtracting  (2)  from  (i), 


log =  2 


^  +  ^' 
3 


^^^...]. . . « 


a  series  involving  only  the  positive  terms  of  series  (i). 

\    '\~  X         7t  ~f"  ft  h 

Putting = ,  we  derive  x  = ;   substituting 

I        X  iz  ^fl  -\-  tl 

in  (3),  v/e  have 

The  Computatio7i  of  Napierian  Logarithfris. 

131.  The  series  given  above  enables  us  to  compute  Napierian 
logarithms.  We  proceed  to  illustrate  by  computing  loge  10. 
The  approximate  numerical  value  of  this  logarithm  could  be 
obtained  by  putting  n  —  i  and  //  =  9  in  (4) ;  but,  since  the  series 
thus  obtained  would  converge  very  slowly,  it  is  more  convenient 
first  to  compute  log  2  by  means  of  the  series  obtained  by  put- 
ting n  z=i  I  and  >^  ~  i  in  (4) ;  thus : 

1  fi    ,    I     I        I    I        I     I  "1 


§  XIX.] 


LOGARITHMTC   SERIES. 


133 


We  then  put  n=  S  and  ^  :=  2  in  (4) ;  whence 


loge  10 


l0ge2    + 


1+i 

-3      3 


1^ 


3^     s   y     7   3""^"  J* 

In  making  the  computation,  it  is  convenient  first  to  obtain 
the  values  of  the  powers  of  ^  which  occur  in  the  series  for  log  2, 
by  successive  division  by  9,  and  afterwards  to  derive  the  values 
of  the  required  terms  of  the  series  by  dividing  these  auxiliary 
numbers  by  i,  3,  5,  /,  etc.  The  same  auxiliary  numbers  are 
also  used  in  the  computation  of  loge  10.  See  the  arrangement 
of  the  numerical  work  below. 


1 
s 

0.3333333333 

I 

0.3333333333 

ar 

370370370 

3 

123456790 

ar 

41 152263 

5 

3230453 

ay 

4572474 

7 

65321I 

ar 

508053 

9 

56450 

ar 

56450 

II 

5132 

ar 

6272 

13 

482 

ar 

697 

15 

46 

ar 

77 

17 

5 

log 

.  2  = 

0.3465735902 
2 

:  0.693  1 47  I  804 

i 

0.3333333333  : 

I 

0.3333333333 

ar 

41 152263  : 

3 

I37I742I 

ar 

508053  : 

5 

ioi6ir 

ar 

6272  : 

7 

896 

ihr 

77  ' 

9 

9 

0.3347153270 
■0.1115717757 

0.2231435513 

Slog 

\2  = 

2.079441 5412 

loge 

jO  = 

2.30258509 

134  THE  DEVELOPMENT  OF  FUNCTIONS.      [Art.    1 3 1. 

The  tabular  logarithms  of  the  system  of  which  lO  is  the 
base,  are  derived  from  the  corresponding  Napierian  logarithms 
by  means  of  the  relation 

loge;r  =  loge  lo  logio^, 

whence  log^o^  =  ^i log^-^  —  M .  loge^- 

logeio  ** 

The  constant  -. ,  denoted  above  by  M.  is  called  the  modulus 

logeio'  ^ 

of  common  logarithms.      Taking  the  reciprocal  of  loggio,  com 

puted  above,  we  have 

M  —  0.43429448. 


The  Developments  of  the  Sine  and  the  Cosine, 
132.  Let  f{x)  =  sin  x, 

then 

f\x)  —  cos  Xyf'\x)  —  —  sin  x,f"'{x)  =  —  cos  x,f''^{pc)  =  sin  ;ir ; 

f'^  being  identical  with/,  it  follows  that  these  functions  recur 
in  cycles  of  four ;  their  values  when  x  —  O  are 

o,  1,0,  —  I,  etc. 

Hence  substituting  in  equation  (i).  Art.  128,  we  have 

x"^       ^         x^  x''  ,  . 

sm  X  =  X +  •  •  •    .   .    (i) 

I-2-3       1-2  •  •  •  5        I-2-  •  •  7  ^  ^ 

In  a  similar  manner,  we  obtain 

x^  X*  x' 

cos  ;ir  =  I + +  .  .  .    .    .    (2) 

1.2       1-2  3-4       1-2  ••  -6  ^  ^ 


§  XIX.]  EXAMPLES.  135 

Examples  XIX. 


/ 


I.  Expand  (i  +  xY- 


(i  +  xr=  I  +  mx  H ^ -X   H ^  ^ -x^  -f  .  .  . 

1-2  I-2-3 

It  is  evident  that  no  coefficient  will  vanish  if  m  is  negative  or  frac- 
tional. This  is  the  form  in  which  the  binomial  theorem  is  employed 
in  computation,  x  being  less  than  unity. 


\/  2.  Find  three  terms  of  the  expansion  of  sin^  x. 

sin^  X  =  x'^ f 

J     3.  Expand  tan  x  to  the  term  involving  x''  inclusive. 

/ 


tan  x^^  X  -\ 1 h 

3  15 


/ 


4.  Expand  sec  x  to  the  term  involving  x^  inclusive. 

x^             <.x^                  dix^ 
sec  ^  =  I  +  —  +  — ^ + 7  + 

1-2  I-2-3-4         1.2  •   •   •   •  6 

5.  Expand  log  sec  ^  to  the  term  involving  x^  inclusive. 


X'  X  X 

log  sec  .^  = 1 \ h 

2         12       45 


/ 

^    6.  Find  four  terms  of  the  expansion  of  £*  sec  x 


J 


o  2X^ 

f'sec^=i  ■\-  X  ■{■  X  -\ h 


7.  Derive  the  expansion  of  log  (i  —  jc')  from  the  logarithmic  series, 
and  verify  by  adding  the  expansions  of  log  {1  ^  x)  and  log  (i  —  x)» 

V     8.  Derive  the  expansion  of  (i  +  ^)f*  from  that  of  f*. 

(i  +  xY'  ^\  ^  2X  ^^- •  •  -h  ■— -x'^, 

^  '  \'2  1-2  ■■  '  n 


136  THE  DEVELOPMENT  OF  FUNCTIONS.    [Ex.    XIX. 

1/    9    Find,  by  means  of  the  exponential  series,  the  expansion  of  xf?'^ 
including  the  /zth  term. 

/  ^ 

V  10.  Expand    — — —    by   division,    making  use    of   the   exponential 

I     "T     <^ 

series. 

^  x^        X^        T.X^        I  \x^ 

y~^—=  I  + +  V". +  •  •  • 
I  +  ^                2         3          8  '       30 
II.  Find  the  expansion  of  f'log(i  +  x)   to  the   term   involving^*, 
by  multiplying  together  a  sufficient  number  of  the  terms  of  the  series 
for  e'  and  for  log  (i  +  x). 


5  ..3  6 


6Mog(i  +  ^)  -^  +  —  +  -  +  ^^-  +  .  .  . 
2         3        40 


v/ 


12.   Expand  log  (i  +  ^'). 


log(i  +  f')  =  log2+-  +  -3--  — + 
13.  Expand  (i  +  s')"  to  the  term  involving  x^  inclusive. 


(i  +£*)«=  2"^!  -h^-x  + 


n{n  +  i)     o(^ 


/ 


j^-v^-ti)    N***           ,    n(7^  +  n  +  2)       x^ 
t.  ^      ^j.    i_  -J- .  - — 


■■} 


14.  Find  the  expansion  of  V{i  ±  sin  2jc),  employing  the  formula 
4^(1  ±  sin  2x)  =  cos  X  ±  sin  x. 

V{i  ±  sin  2x)  =  I  ±x  -  -—  =F  -^   +  •  •  • 

1-2        1.2.3 

15.  Find    the    expansion    of    cos"  j^t:    by    means    of    the    formula 
cos'^.x:  =  J(i  4-  cos  2x). 

,  2  2'X*  2\x'' 

cos'^  =  I  —  ^^  H f-  .  .  . 

I-2-3-4       1-2  ...  6 


§  XIX.]  EXAMPLES.  137 

i/  16.  Find   the    expansion    of    cos'  x^    by   means    of    the    formula 
cos'  X  =  J(cos  sx  +  $  cos  x). 


1*2  4         I-2-3-4  4  1-2  •  •  •  2« 

A''^      17.  Compute  loge3,   and  find  log;o3  by  multiplying  by  the  value  of 
il/  (Art.  166). 


18.  Find  loge269. 

J*uf  n  —  270  =  10  X  3",  and /i  =  —  i. 

19.  Find  log,  7,  and  log.  13. 


loge3  =  1. 0986 1 23. 
Iogio3  =  0.477 1 2 13. 


loge269  =:  5.5947  1 14. 

I0ge7   =    I.9459IOI. 
l0geI3  =  2.5649494. 


CHAPTER   VIII. 
Curve  Tracing. 


XX. 

Equations  in  the  Form  y  =  f(x). 

133.  When  a  curve  given  by  its  equation  is  to  be  traced, 
it  is  necessary  to  determine  its  general  form  especially  at  such 
points  as  present  any  peculiarity,  and  also  the  nature  of  those 
branches  of  the  curve,  if  there  be  any,  which  are  unlimited  in 
extent. 

The  general  mode  of  procedure,  when  the  equation  can  be 
put  in  either  of  the  forms,  y  =f{x)  ox  x  =  #(/),  is  indicated  in 
the  following  examples. 

Asymptotes  Parallel  to  the  Coordinate  Axes, 

(34.  Example  \.  -ay  —  xy=a'' (i) 

Solving  for  J/,  we  obtain 

^  =  ^:^' •   •    •   (2) 

WhGnx  =  o,y  =  a.  Numerically  equal  positive  and  nega- 
tive values  of  x  give  the  same  values  for  jj/;  the  curve  is  there- 
fore symmetrical  with  reference  to  the  axis  of  y.    As  x  increases 


§  XX.]         ASYMPTOTES  PARALLEL    TO    THE  AXES. 


139 


from  zero,  y  increases  until  the  denominator,  a^  —  x^,  becomes 
zero,  when  y  becomes  mfinite  ;  this  occurs  when  x  =  ±  a. 

Draw  the  straight  Hnes  x  =  ±  a.  These  are  hnes  to  which 
the  curve  approaches  indefinitely,  for  we  may  assign  values  to 
X  as  near  as  we  please  to  -\-  a  ov  to  —  a,  thus  determining  points 
of  the  curve  as  near  as  we  please  to  the  straight  lines  x=  a  and 
X  =  —  a.     Such  lines  are  called  asymptotes  to  the  curve. 

When  X  passes  the  value  a,y  becomes 
negative  and  decreases  numerically,  ap- 
proaching the  value  zero  as  x  increases 
indefinitely.  Hence  there  is  a  branch 
of  the  curve  below  the  axis  of  x  to 
which  the  lines  x  =  a  and  y  =^  o  are 
asymptotes. 

The  general  form  of  the  curve  is  in- 
dicated in  Fig.  15. 

The  point  (o,  a)  evidently  corresponds  to  a  minimum  ordi- 
nate. 


Fig.  15. 


135.   Example  2.     a^x  =y  {x  —  of (i) 

Solving  for^, 


y  = 


{X  -  a)^ 


(2) 


When  X  is  zero,  y  is  zero ;  y  increases  as  x  increases  until 
X  =  ay  when  y  becomes  infinite.     Hence 


X  =  a 


is  the  equation  of  an  asymptote.  When  x  passes  the  value 
<2,  y  does  not  change  sign,  but  remains  positive,  and  as  x  in- 
creases y  diminishes,  approaching  zero  as  x  increases  indefi- 
nitely. 


I40  CURVE    TRACING.  [Art.    1 35. 

Examining  now  the  values  of  j  which  correspond  to  nega- 
tive values  of  x^  we  perceive  that,  y  becoming 
negative,  the  branch  which  passes  through  the 
origin  continues  below  the  axis  of  x,  and  that 
y  approaches  zero  as  the  negative  value  of  x 
Fig.  17.  increases  indefinitely.     Hence  the  general  form 

of  the  curve  is  that  indicated  in  Fig.  17. 

(36.  To  determine  the  direction  of  the  curve  at  any  point, 
we  have 

,      dy  ^    a  4-  X  ,  . 

The  direction  in  which  the  curve  passes  through  the  origin 
IS  given  by  the  value  of  tan  (/>  which  corresponds  to  x  =  o. 
From  (3),  we  have 


^^Jo 


dx_ 
the  inclination  of  the  curve  at  the  origin  is  therefore  45'' 

Minimum  Ordinates  and  Points  of  Inflexion. 


(37.  To  find  the  minimum  ordinate  which  evidently  exists 
he  left  of  the  ax 
to  zero,  and  deduce 


dy 
on  the  left  of  the  axis  of  j,  we  put  the  expression  for  -^  equal 


x  —  —  a. 


The  minimum  ordinate  is  therefore  found  at  the  point  whose 
abscissa  is    —  ^  ;    its   value,    obtained   from    equation    (2),   is 

d'^v 
K  point  of  inflexion  is  a  point  at  which  ^,  changes  sign  (see 


§  XX.]  POINTS  OF  INFLEXION.  I4I 

Art.  74) ;  in  other  words,  it  is  a  point  at  which  tan  ^  has  a 
maximum  or  a  minimum  value.  In  this  case  there  is  evidently 
a  point  of  inflexion  on  the  left  of  the  minimum  ordinate.  From 
equation  (3)  we  derive 


dx'      "     {x-aY 

putting  this  expression  equal  to  zero  to  determine  the  abscissa, 
and  deducing  the  corresponding  ordinate  from  (2),  we  obtain, 
for  the  coordinates  of  the  point  of  inflexion, 

X  =  —  2a^    and    y  —  —\a. 


Oblique  Asymptotes, 

\ZZ,  Example  I.  x' —  2A:y  —  2x' —Sy  =  O,  c  ..  .  ,  (l) 
Solving  this  equation  for  j,  we  have 

^=J'x^-T4     ••     —     -•.    (2) 

It  is  obvious  from  the  form  of  equation  (2)  that  the  curve 
meets  the  axis  of  x  at  the  two  points  (o,  o)  and  (2,  o).  Since 
y  is  positive  only  when  ;r  >  2,  the  curve  lies  below  the  axis  of 
X  on  the  left  of  the  origin,  and  also  between  the  origin  and  the 
point  (2,  o),  but  on  the  right  of  this  point  the  curve  lies  above 
the  axis  of  x. 

139  Developing  the  second  member  of  equation  (2)  into 
an  expression  involving  a  fraction  whose  numerator  is  lower  in 
degree  than  its  denominator,  we  have 

,=  J^-I+2^,^.      .      .      ...     (3) 


142 


CURVE    TRACING. 


[Art.  139. 


The  fraction  in  this  expression  decreases  without  limit  as  x  in- 
creases indefinitely  ;  hence  the  ordinate  of  the  curve  may,  by 
increasing  jc,  be  made  to  differ  as  little  as  we  please  from  that 
of  the  straight  line 


This  line  is,  therefore,  an  asymptote. 


The    fraction 


2  —  X 


X'  +  4 
is  positive  for  all  values 
of  X  less  than  2,  negative 
for  all  values  of  x  greater 
than  2,  and  does  not  be- 
come infinite.  The  curve, 
therefore,  lies  above  the 
F^^"  ^^'  asymptote  on  the  left  of 

the  point  (2,  o),  and  below  it  on  the  right  of  this  point,  as 

represented  in  Fig.  18. 


14-0.  There  is  evidently  a  minimum  ordinate  between  the 
origin  and  the  point  (2,  o). 

We  obtain  from  equation  (2) 


and 


df  _ 
dx  ~ 

d^  _ 

dx"  ~ 


x^  ^  \2x 


16 


vi''4-  (yx^-\-  \2x  — 


(^'  +  4)' 


(4) 
(5) 


dy  _ 


Putting  -j^  =  o,  we  obtain  x  —  O  and  x  =^  1.19  nearly,  the  only 

real  roots;  the  abscissa  corresponding  to  the  minimum  ordinate 
is  therefore  1.19,  the  value  of  the  ordinate  being  about  —  o.ii. 
The  root  zero  corresponds  to  a  maximum  ordinate  at  the 
origin. 


§XX.] 


OBLIQUE  ASYMPTOTES. 


143 


Putting 


(Pi 


O,   we  obtain   the  three   roots  ;ir  =  —  2,  and 


4r  =  2  (2  ±  V  3)  ;  the  corresponding  ordinates  are  obtained 
from  equation  (3).  There  are,  therefore,  three  points  of  in- 
flexion, one  situated  at  the  point  (—  2,  —  i),  and  the  others 
near  the  points  (0.54,  —  0.05),  and  (7.46,  2.55). 

The  inchnation  of  the  curve  is  determined  by  means  of 
equation  (4)  to  be  tan- 'J  at  the  point  (2,  o),  and  tan"' J  at  the 
left-hand  point  of  inflexion. 


(41.   Example  4.     x"  —  xy  ^ 

Solving  for/ 

.r'  +  I 
^= — :::— 


Curvilinear  Asymptotes, 
=-  o.    ,    , 


=  x'  -\- 


(0 
(2) 


In  this  case,  on  developing  jj/  in  powers  of  x,  the  integral  portion 
of  its  value  is  found  to  contain  the  second  power  of  x ;  the 
fraction  approaches  zero  when  x  increases  indefinitely  ;  hence 
the  ordinate  of  this  curve  may  be  made  to  differ  as  little  as  we 
please  from  that  of  the  curve 

j^  =  x' (3) 

The  parabola  represented  by  this  equation 
is  accordingly  said  to  be  a  curvilinear  asymp- 
tote. It  is  indicated  by  the  dotted  line  in 
19. 


Fig 


142.   The  sign  of  the  fraction  -  is  always 

the  same  as  that  of  x,  and  its  value  is  infinite 
when  X  is  zero ;  hence  the  curve  lies  below 
the  parabola  on  the  left  of  the  axis  of  7,  and 
above  it  on  the  right,  this  axis  being  an 
asymptote,  as  indicated  in  Fig.  19. 


Fig.  19. 


144  CURVE    TRACING.  [Art.    1 42. 

Taking  derivatives,  we  obtain 

^  =  ^^~^ (4) 

and  g=.(n.i,).     ......     (5) 

There  is  a  point  of  inflexion  at  (  —  i,  o) ;  the  inclination  of  the 
curve  to  the  axis  of  x  at  this  point  is  tan  ~'  (—  3). 

There  is  a  minimum  ordinate  at  the  point  (^^4,  \^2). 

This  cubic  curve  is  an  example  of  the  species  called  by- 
Newton  the  trident.  The  characteristic  property  of  a  trident 
is  the  possession  of  a  parabolic  asymptote  and  a  rectilinear 
asymptote  parallel  to  the  axis  of  the  parabola. 


Examples  XXVI. 

J      I.  Trace  the  curve  j  ==  x  (^^  —  i). 

Since  J  is  an  odd  function  of  x^  the  curve  is  symmetrical  with  re- 
ference to  the  origin  as  a  centre.  Find  the  point  of  inflexion,  and 
the  minimum  ordinate. 

2.  Trace  the  curve  y  (jr  —  i )  =  ^'. 

The  curve  has  for  an  asymptote  the  line  :r  =  i  ;  there  is  a  mini- 
mum ordinate  at  (2,  2),  and  a  point  of  inflexion  at  (4,  |  i/3). 

>l  3.  Trace  the  curve  y"  —  x"  {x  —  a),  determining  its  direction  at 
the  points  at  which  it  meets  the  axis  of  x. 

The  asymptote  is  found  by  the  method  of  development,  thus 

the  equation  of  the  asymptote  is  therefore 


§  XX.]  EXAMPLES.  145 

4.  Trace  the  curve  x^  ^  xy  ^-  2x  —  y  =  o. 

5.  Trace  the  curve  y"  =  x"  -\-  x^^  and  find  its  direction  at  the 
origin. 

The  curve  has  a  maximum  ordinate  at  (— |,  ±  f  1/3).  The 
value  of  y  may  be  taken  as  the  function  whose  maximum  is  required. 
(See  Art.  108.) 

6.  Trace  the  curve  j;  =  ^'  —  xy.  Find  the  point  of  inflexion,  the 
minimum  ordinate,  and  the  asymptotes. 

The  curve  has  a  rectihnear  asymptote  x—  —  i,  and  a  curvihnear 
asymptote  ji'  =^"^  —  ^  4-  i.     This  curve  is  a  trident.     (See  Art.  142.) 

7.  Trace  the  curve  y^  =  jc^  —  x^. 

Both  branches  of  the  curve  are  tangent  to  the  axis  of  x  at  the 
origin. 

8.  Trace  the  curve  jv'  —/^y  =  x*  +  y^. 
Solving  for^,  we  obtain 

y^2  ±   s/{x^  +  ^'  +  4)  =  2  ±  i/[(jc  +  2)  (x'  -  ^  +  2)]. 

The  factor  x^  —  x  -^  2  being  always  positive,  the  curve  is  real  on 
the  right  of  the  line  x  =  —  2. 

Find  the  points  at  which  the  curve  cuts  the  axis,  and  show  that 
the  upper  branch  has  a  maximum  ordinate  for  ::c  =  —  f  and  a  mini- 
mum ordinate  for  x  —  o. 

9.  Trace  the  curve  {x  —  2a)  xy  =  a(x  —  a)  (x  —  3^). 

10.  Trace  the  curve  {x  —  2a)  xy-  =  a"  {x  —  a)  (x  —  ^a). 

11.  Trace  the  curve/  =  ^*  (i  —  jv*)' :  find  all  the  points  at  which 
the  tangent  is  parallel  to  the  axis  of  x. 

12.  Trace  the  curve  6x  (i  —  x)  y  =  i  +  3^. 

This  curve  has  a  point  of  inflexion,  determined  by  a  cubic  having 
only  one  real  root,  which  is  between  —  i  and  —  2.  Find  the  three 
asymptotes,  and  the  maximum  and  the  minimum  ordinate. 


146  CURVE    TRACING.  [Ex.    XX. 

13.  Trace  the  curve  ^y  —  {x  -^  2f  {1  •\-  x^). 
Solving  the  equation  for  j',  we  have 


y=± 


^4/(i+^')=±(2+.r)(i+i^y. 


The  asymptotes  are  jj;  =  a;  +  2,  j  =  —  .r  —  2,  and  x  =0.  The 
curve  has  a  minimum  ordinate  corresponding  to  a  =  .y/2  ;  the  inclina- 
tion at  the  point  at  which  it  cuts  the  axis  of  x  is  tan"'  (±^1/5).  There 
is  a  point  of  inflexion  corresponding  to  the  abscissa  x  =  —  6.1  nearly. 


XXL 

Curves  Given  by  Polar  Equations, 

143.  The  following  examples  will  illustrate  some  of  the 
methods  employed  when  the  curve  is  given  by  means  of  its 
polar  equation. 

Example  t^,     r  =  acosd cos  26.      .     .      .      .      ,     .     .     .     (i) 

When  6  =  o^r  —  a^  the  generating  point  P  therefore  starts 
from  A  on  the  initial  line.  As  B  increases,  r  decreases  and 
becomes  zero  when  ^  =  45°, /*  describing  the  half-loop  in  the 
first  quadrant,  and  arriving  at  the  pole  in  a  direction  having  an 
inclination  of  45°  to  the  initial  line.  When  8  passes  45°,  r 
becomes  negative,  and  returns  to  zero  again  when  6  —  90°,  P 
describing  the  loop  in  the  third  quad- 
rant. As  6  passes  90°,  r  again  becomes 
positive,  but  returns  to  zero  when 
0=  135°,  P  describing  the  loop  in  the 
second  quadrant.  As  Ovaries  from  135'' 
to  180°,  r  again  becomes  negative,  /^de- 
^^*  scribing  the  half-loop  in  the  fourth  quad- 

rant, and  returning  to  ^, 


g  XXL]        CURVES  GIVEN  BY  POLAR  EQUATIONS.  I47 

In  this  example  if  we  suppose  d  to  vary  from  180°  to  360°, 
P  will  again  describe  the  same  curve,  and,  since  d  enters  the 
equation  of  the  curve,  by  means  of  trigonometrical  functions 
only,  it  is  unnecessary  to  consider  values  of  6  greater  than  360°. 

144,  Putting  equation  (i)  in  the  form 

r  —  a{2  cos^O  —  cos  6), 


we  derive 


^  =  a{-6 cos^^ sin  6  +  sin  6). 
du 


To  determine  the  maxima  values  of  r,  we  place  this  derivative 
equal  to  zero,  thus  obtaining  the  roots 

sin  ^  =  o     and     cos  Q  =  ±\  V6', 

the  former  gives  the  point  A  on  the  initial  line,  and  the  latter 
gives  the  values  of  0  which  determine  the  position  of  the  maxi- 
ma in  the  small  loops.  The  corresponding  values  of  r  are  =F  -  V6. 

To  determine  the  position  of  the  maximum  ordinate,  we 
have  from  (i) 

y  =  rsln  0  =  ia  sin  4^. 

The  maxima  values  occur  when  sin  4^9  =1,  and  the  minima 
when  sin  4^  =  —  I  ;  that  is,  we  have  maxima  when  0  —  Itt  and 
when  6  =  ^7ty  and  minima  when  6  =  |7r  and  -}7t. 

145.  In  the  preceding  example  the  substitution  of  ^  +  tt 
for  6  changes  the  sign  but  not  the  numerical  value  of  r.  When 
this  is  the  case,  ^and  6  +7r  evidently  give  the  same  point  of 
the  curve,  and  the  complete  curve  is  described  while  0  varies 
from  o  to  7t.  If  however  this  substitution  changes  neither  the 
numerical  value  nor  the  sign  of  r,  ^and  0  -h  7t  will  give  points 
symmetrically  situated  with  reference  to  the  pole ;  that  is,  the 
curve  will  be  symmetrical  in  opposite  quadrants. 


148  CURVE    TRACING.  [Art.  1 45. 

Again  if  the  substitution  of  —  ^  for  Q  does  not  change  the 
value  of  r,  6  and  —  B  give  points  symmetrically  situated  with 
reference  to  the  initial  line,  hence  in  this  case  the  curve  is  sym- 
metrical to  this  line ;  but,  if  the  substitution  of  —  ^  for  6 
changes  the  sign  of  r  without  changing  its  numerical  value,  the 
curve  is  symmetrical  with  reference  to  a  perpendicular  to  the 
initial  line. 


The  Determination   of  Asymptotes   by  Means  of  Polar 

Equations, 

(46.  When  r  becomes  infinite  for  a  particular  value  of  d  the 
curve  has  an  infinite  branch,  and,  if  there  be  a  corresponding 
asymptote,  it  may  be  determined  by  means  of  the  expression 
derived  below. 

Let  By  denote  a  value  of  d  for  which  r  is  infinite,  and  let  OB 
be  drawn  through  the  pole,  making  this  angle  with 
the  initial  line ;  then,  from  the  triangle  OBP,  Fig. 
20,  we  have 


PB^  rsm{d,-d). 


Fig.  20. 


Now,  if  the  curve  has  an  asymptote  parallel  to 
OB,  it  is  plain  that  as  6  approaches  6^  the  limiting  value  of  PB 
will  be  equal  to  OR,  the  perpendicular  from  the  pole  upon 
the  asymptote.  Hence,  if  the  curve  has  an  asymptote  in  the 
direction  6^,  the  expression 

OR=\rsm{d,  -  e)\, 

which  takes  the  form  o^  •  o,  will  have  a  finite  value,  and  this 
value  will  determine  the  distance  of  the  asymptote  from  the 
pole.  Fig.  20  shows  that  when  the  above  expression  is  posi- 
tive OR  is  to  be  laid  off  in  the  direction  6^  —  90°. 

If  upon  evaluation  the  expression  for  OR  is  found  to  be  in- 


§XXI.] 


ASYMPTOTES. 


149 


finite   we  infer  that  the   infinite  branch   of  the   curve  is  para- 
bolic. 


147.  Example  6. 


r  — 


aO" 


(I) 


Since  r  becomes  infinite  when  6  =  i,  we  proceed  to  apply  the 
method  established  in  the  preceding  article  for  determining  the 
existence  of  an  asymptote.     In  this  case  we  have 


[r  sin  {e^  -  6)] 


_  r  aO'      sin(i  -  <9)-j  _ 


The  angle  0  =  i  corresponds  to  57°  18',  nearly,  and  since 
the  expression  for  the  perpendicular  on  the  asymptote  is  neg- 
ative its  direction  is  ^j  +  90°  =  147°  18';  consequently,  the 
asymptote  is  laid  off  as  in  Fig.  21. 

Numerically  equal  positive  and  negative  values  of  6  give  the 
same  values  for  r ;  hence  the  curve  is  symmetrical  with  refer- 
ence to  the  initial  line. 

While  0  varies  from  o  to  i,  r  is  negative  and  varies  from  o 
to  00,  giving  the  infinite  branch  in  the  third 
quadrant. 

As  d  passes  the  value  unity,  and  increases 
indefinitely,  r  becomes  positive  and  decreases, 
approaching  indefinitely  to  the  limiting  value 
a,  which  we  obtain  from  (i)  by  making  6  in- 
finite. Hence  the  curve  describes  an  infinite 
number  of  whorls  approaching  indefinitely  to 
the  circle  r  —  a^  which  is  therefore  called  an 
asymptotic  circle. 

The  points  of  inflexion  in  this  curve  are 
determined  in  Art.  175. 


Fig.  ±i. 


I50  CURVE    TRACING.  [Ex.  XXI. 


i 


Examples  XXI. 


I.  Trace  the  curve  r^=^  a  cos"  \  0. 

Show  that,  to  describe  the  curve,  0  must  vary  from  o  to  3  tt  ;  also 
that  the  curve  is  symmetrical  to  the  initial  line.  Find  the  values  of  0 
which  correspond  to  the  maxima  and  minima  ordinate*  and  abscissas, 
the  initial  line  being  taken  as  the  axis  of  x. 


\l 


2.  Trace  the  curve  r  =  a  {2  s'lnO  —  ^  sin'o). 

Show  that  the  entire  curve  is  described  while  0  varies  from  o  to  tt^ 
and  that  the  curve  is  symmetrical  with  reference  to  a  perpendicular 
to^the  initial  line. 


7 


3.  Trace  the  curve  r  =  2  +  sm  30. 

A  maximum  value  of  r  (equal  to  3)  occurs  at  0  =  30°  ;  a  mini- 
mum (equal  to  i)  at  0  =  90°.  The  curve  is  symmetrical  with  refer- 
ence to  lines  inclined  at  the  angles  30°,  90°,  and  150°  to  the  initial 
line. 


nI 


J 


4.  Trace  the  curve  r  =  i  4-  sin  50. 
The  curve  consists  of  five  equal  loops. 

5.  Trace  the  curve  r"  =  a^  sin  30. 

The  curve  consists  of  three  equal  loops. 


/    6.  Trace  the  curve  r  cos  0  =  a  cos  2O, 

The  curve  has  an  asymptote  perpendicular  to  the  initial  line  at  the 
distance  a  on  the  left  of  the  pole. 

V   7.  Trace  the  curve  r  =  2  +  sin  |0. 

A  m.aximum  value  of  r  occurs  at  0  =  60°,  and  a  minimum  at 
0  =  180°.  The  curve  has  three  double  points,  one  being  on  the  initial 
line. 

n     8.  Trace  the  curve  r  cos  20  =  a. 

The  curve  is  symmetrical  with  reference  to  the  initial  line  and 
with  reference  to  a  perpendicular  to  the  initial  line.  There  are  four 
asymptotes. 


§  X X I .]  EXAMPLES.  1 5 1 

9.  Trace  the  curve  r  sin  40  =  «  sin  t^^. 

The  curve  is  symmetrical  to  the  initial  line,  and  has  three  asymp- 
totes ;  the  minimum  value  of  r  is  \a. 

10.  Trace  the  curve  r"  =  c^  cos  26. 

The  curve  is  symmetrical  with  respect  to  the  pole  since 
r  =.  -^a  *^  (cos  20)  :  r  is  imaginary  for  values  of  0  between  \Tt  and  J^. 

3.         3. 

11.  Trace  the  curve  r*  =  a^  cos  |9. 

The  curve  consists  of  three  equal  loops,  r  being  real  for  all  values 
of  e. 

12.  Trace  the  curve  r^  cos  Q  =  ^'^  sin  2i^. 

The  curve  consists  of  two  loops  and  an  infinite  branch  which  has 
an  asymptote  perpendicular  to  the  initial  line  and  passing  through  the 
pole. 

29 

i^.  Trace  the  curve  r  =  ^  . 

^  20    —    1 

Find  the  rectilinear  and  the  circular  asymptote,  and  also  the  point 
of  inflexion. 


XXII. 

The  Parabola  of  thejwth  Degree. 

(48.  The  term  parabola  is  frequently  applied  to  any  curve 
in  which  one  of  the  coordinates  is  proportional  to  the  n\}ix 
power  of  the  other,  n  being  greater  than  unity.  The  parabola 
proper  is  thus  distinguished  as  the  parabola  of  the  second 
degree. 

The  general  equation  of  the  parabola  of  the  ;^th  degree  is 
usually  written  in  the  homogeneous  form,  {a  being  positive) 

^n-r y  —  x"" , 


152 


CURVE    TRACING. 


[Art.   148. 


The  curve  passes  through  the  origin  and  through  the  point 
{a^  a),  for  all  values  of  n.  Since  7i  >i,  the  curve  is  tangent  to 
the  axis  of  x  at  the  origin. 

(49.'  The  following  three  diagrams  represent  forms  which 
the  curve  takes  for  different  values  of  n.  When  n  denotes  a 
fraction,  it  is  supposed  to  be  reduced  to  its  lowest  terms. 

Fig.  22  represents  the  general  shape  of 
the  curve  when  n  is  an  even  integer,  or  a 
fraction  having  an  even  numerator  and  an 
odd  denominator. 

Fig.  23  represents  the  form  of  the  curve 
when  11  is  an  odd  integer  or  a  fraction  with 
an  odd  numerator  and  an  odd  denominator, 


Fig.  22. 


the  origin  being  a  point  of  inflexion. 

Fig.  24  represents  the  form  of  the 
curve  when  /^  is  a  fraction  having  an  odd 
numerator  and  an  even  denominator. 
In  this  case  y  is  regarded  as  a  two- 
valued  function,  and  is  imaginary  when 


Fig.  23. 
Fig.  22  is  constructed  for  the  parabola 
in  which  ;2  =  4. 

Fig.  23  is  the  cubical  parabola  in  which 

2  Fig.  24  is  the  semi-cubical  parabola  in 

which  n  =  ^  ;  the  equation  being 


Fig.  24, 


or 


^2jJ/ 


ay 


,      1 


The  curves  corresponding  to  the  general  equation 
y=A-^Bx+Cx'^  Dx'+  .  .  ,  Lx" 
are  sometimes  cdiWed  parabolic  curves  of  the  nih.  degree. 


§  XXIL] 


THE   CISSOID   OF  DIOCLES. 


153 


The  Cissoid  of  Diodes, 

ISO.  Let  ^  be  a  point  on  the  circumference  of  a  circle, 
and  BC  a  tangent  at  the  opposite 
extremity  of  the  diameter  AB\  let 
AC  be  any  straight  line  through  A^ 
and  take  CP  =  AD ;  then  the  locus 
of  P  is  the  cissoid. 

To  find  the  polar  equation,  AB 
being  the  initial  line,  let  DB  be 
drawn,  and  denote  the  radius  of  the 
circle  by  a\  then  AC  —2a  ^ecO \ 
and  since  ADB  is  a  right  angle, 
AD  —  2a  cos  8.  The  polar  equation 
of  the  locus  of  P^  A  being  the  pole, 
is,  therefore, 


r  =  2^  (sec  ^— cos  &) 


cos  6 


or 


2a 


sm' 


cos  0 


(I) 


151.  To  obtain  the    rectangular  equation,  we  employ  the 
equations  of  transformation 


sin  6  = 


-.y 


cos  c/  =  — , 

r 


r''  =  x'-Vf\ 


whence,  eliminating  6  we  obtain 


y 

r  —  2a  ^^, 
rx 


and  thence  the  rectangular  equation  of  the  curve 

xix"  ^  f)  =  2af,     .     .     . 


or 


/  = 


2a  —  X 


»!♦?♦•• 


(2) 

(3) 


154 


CURVE    TRACING. 


[Art.  152. 


The  Cardioid. 
152.  The  curve  defined  by  the  polar  equation 

r  =  2^(1  --cos  d)  .    .    (i) 

is  called   the  cardioid.     In  Fig. 
26,  A  denotes  the  pole. 

The  polar  equation  can  also 
be  written  in  the  form 


Fig.  26. 


r  =  4<2;  sin^  \Q 


(2) 


Transforming  equation  (i)  to 
rectangular  coordinates,  we  have 
for  the  rectangular  equation  of 
the  cardioid, 


{x"  +  /)2  +  A^x  {p^  +  /)  -  4^y  =  o 


(3) 


A  point  at  which  two  branches  of  a  curve  have  a  common 
tangent  is  called  a  cusp.     This  curve  has  a  cusp  at  the  origin. 


The  Lemniscata  of  Bernoulli, 
153.  The  curve  defined  by  the  polar  equation 
^2  =  ^2  cos  2^     .     .     .     (i) 

is  called  the  lemniscata.  In  Fig.  27 
O  denotes  the  pole  :  a  is  the  semi- 
axis  of  the  curve. 

From  (i),  we  have  Fig.  27. 


r^  =  «2(cos2^-sin2  6^), 


§  XXII.]  THE  LEMNISCATA.  155 

or  r'^a^ — ^  ; 

whence  we  have 

{x"  ^ff^a\f-x')^o, (2) 

the  rectangular  equation  of  the  lemniscata,  referred  to  its  cen- 
tre and  axis  of  symmetry. 

If  we  turn  the  initial  line  back  through  45°,  (i)  becomes 

7-2  =  «2  gjj^  2^, (3) 

and  the  corresponding  rectangular  equation  is 

(;r2+//  =  2^;rj/ (4) 

When  the  equation  has  this  form,  the  coordinate  axes  are  the 
tangents  at  the  origin. 

The  Logarithmic  or  Equiangular  Spiral, 

(54.  This   spiral    is   defined   by  the 
polar  equation 


1 


...  (I) 

or  log  r  —  log  a  +  nO^ 

the  logarithm  of  the  radius  vector  being  ^lo,  28 

a  linear  function  of  the  vectorial  angle. 

It  is  proved  in  Art.  168  that  this  curve  cuts  its  radius  vector 
at  a  constant  angle  whose  cotangent  is  n ;  hence  it  is  sometimes 
called  the  equiangular  spiral. 

The  Loxodromic  Curve. 

(55.  The  track  of  a  ship  whose  course  is  uniform  is  a  curve 
that  cuts  the  meridians  of  the  sphere  at  a  constant  angle,  and 
is  called  a  loxodromic  curve. 


156  CURVE    TRACING.  [Art.    1 5 5. 

If  we  project  this  curve  stereographically  upon  the  plane  of 
the  equator  the  meridians  will  project  into  straight  lines,  and, 
since  in  this  projection  angles  are  unchanged  in  magnitude,  the 
projection  of  the  curve  will  make  a  constant  angle  with  the 
projections  of  the  meridians  and  will  therefore  be  an  equiangu- 
lar spiral. 

Let  d  denote  the  longitude  of  the  generating  point  mea- 
sured from  the  point  at  which  the  curve  cuts  the  equator,  and 
C  the  course ;  that  is,  the  constant  acute  angle  at  which  the 
curve  cuts  the  meridians,  the  generating  point  being  supposed 
to  approach  the  pole  as  6  increases.  Taking  as  the  pole  the 
projection  of  the  pole  of  the  sphere,  the  polar  equation  of  the 
projected  curve  will  be  of  the  form 

r  =  ^^«^ (i) 

in  which  a  is  the  radius  of  the  sphere,  since  ^  =  o  gives  r  ^=  a\ 
we  also  have 

n  =  —  cot  C, (2) 

since  the  angle  whose   cotangent  is  ;/  is  the  supplement  of  C 
(see  the  preceding  article). 

Denoting  by  ^  the  co-latitude  of  the  projected  point  we 
have,  by  the  mode  of  projection, 

-  =  tan  i^  ;         (3) 

a 

and,  denoting  the  corresponding  latitude  by  /, 

Equation  (i)  is  therefore  equivalent  to 

tana7r-^/)  =  f-^-*^; 
whence,  solving  for  6^  we  have 


XXII.] 


THE  LOXODROMIC  CURVE. 


157 


(9  =  -  tan  (T  loge  tan  (Itt  -  \l)  =  tan  C  loge  tan  (jTr  +  |/), 
or,  employing  common  logarithms  and  expressing  (9  in  degrees, 
e°  =  131.9284  tan  C  logxo  tan  (45°  +  1/)  •     .     .     (4) 


T/^e  Cycloid, 

156.  The  path  de- 
scribed by  a  point  in 
the  circumference  of  a 
circle  which  rolls  upon 
a  straight  line  is  called 
a  cycloid.  The  curve 
consists  of  an  unlimited 
number  of  branches  cor- 
responding to  successive  revolutions  of  the  generating  circle  ;  a 
single  branch  is,  however,  usually  termed  a  cycloid. 

Let  O^  the  point  where  the  curve  meets  the  straight  line, 
be  taken  as  the  origin,  let  P  be  the  generating  point  of  the 
curve,  and  denote  the  angle  PCR  by  ^.  Since  PR  is  equal  to 
the  line  OR  over  which  it  has  rolled, 

0R  =  PR  =  aip, 

and,  from  Fig.  29,  we  readily  derive 


X  =  a(ip  —  sin  f)   1 
y  =  a  {i  —  cos  Jp)   J 


(0 


157.  These  two  equations  express  the  values  of  x  and  f  in 
terms  of  the  auxiliary  variable  «/?,  and  constitute  the  equations 
of  the  cycloid.  If  desirable,  ?/?  is  easily  eliminated  from  equa- 
tions (i)  and  an  equation  between  ;i;  and  /  obtained.  Thus, 
from  the  second  equation,  we  have 


158  CURVE    TRACING,  [Art.    1 57. 

COS  ^  =  ^^y  ^     whence     sin  th  =  n^^7~j) 
a  a 

and  hence  from  the  first  of  equations  (i) 

;r=^cos-' ^  —  ^{2ay  —  y"),     ,     .     .     .     (2) 

or  x=  a  vers-' -  —  1/(2^7  —  y)- 

Equations  (i)  will  in  general  be  found  more  convenient  than 
equation  (2).     Thus  we  easily  derive  from  (i) 

dy  ^        s'mtp  dip        _      sin  tfj 
dx      (i  —  cos tp)  dtp  ~  I  —  costp* 
whence 

dy  _   d  fdy  \  _    cos  ^p  —  \    dip  _  _  i 

dx^  ~~  dx  \dx  J       (i  —  cos  ipy  dx~        a{\  —  cos  r^^y  ' 

.  158.  The  cycloid  is  frequently  referred  to  the  middle  point 
O'  or  vertex  of  the  curve  as  an  origin,  the  directions  of  the 
axes  being  turned  through  90°. 

Denoting  the  coordinates  referred  to  the  axes  O'X'  and 
O'Vy  in  Fig.  29,  by  x'  and/',  we  have 

y  =  X  —  art  =  a  {ip  —  TT  —  sin  tp), 
x'  =  2a  —  y  —  a{i  +  cos  ^-?), 

or,  denoting  ^  —  tt  by  ip\ 


y  =  a  (tp''  +  sin  tp') 
x'=  a{i    —  cos  tp') 


(3) 


In  these  equations  tp'  =  o  gives  the  coordinates  of  the  ver- 
tex and  tp'  =  ±  Tt  gives  those  of  the  cusps. 


§  XXII.] 


THE   EPICYCLOID. 


159 


The  Epicycloid, 


159.  When  a  circle,  tan- 
gent to  a  fixed  circle  exter- 
nally, rolls  upon  it,  the  path 
described  by  a  point  in  the 
circumference  of  the  rolling 
circle  is  called  an  epicycloid. 
Taking  the  origin  at  the 
centre  of  the  fixed  circle, 
and  the  axis  of  x  passing 
through  A,  (one  of  the  posi- 
tions of  P  when  in  contact 
with  the  fixed  circle,)  a,  by  rp, 
and  Xy  being  defined  by  the 
diagram,  we  have,  evidently, 


Fig.  30. 


a^p  =  bx  .'.  X  =  -^  ^. 


The  inclination  of  CP  to  the  axis  of  x  is  equal  to  ^  +  j,  or  to 
— 7 — ip  ;  the  coordinates  of  /'are  found  by  suotracting  the  pro- 
jections of  CP  on  the  axes  from  the  corresponding  projections 
of  OC',  hence 


X  =  {a  +  b)  cos  ip  -  b  cos  — - —  ?/? 
y  =  {a  +  b)  smip  —  b  sin  — - —  ip 


(0 


These  are  the  equations  of  an  epicycloid  referred  to  an  axis 
passing  through  one  of  the  cusps. 


i6o 


CURVE    TRACING. 


[Art.  159. 


Were  the  generating  point  taken  at  the  opposite  extremity 
of  a  diameter  passing  through  P  in  the  figure,  the  projection  of 
6P  would  be  added  to  that  of  0C\  the  axis  of  x  would  in  this 
case  pass  through  one  of  the  vertices  of  the  curve,  and  the 
second  terms  in  the  above  values  of  x  and  y  would  have  the 
positive  sign. 


The  Hypocycloid. 


Fig.  31. 


160.  When  the  rolling 
circle  has  internal  contact 
with  the  fixed  circle,  the 
curve  generated  by  a  point 
on  the  circumference  is  called 
the  hypocycloid^  whether  the 
radius  of  the  rolling  circle  be 
greater  or  less  than  that  of 
the  fixed  circle. 

Adopting  the  notation 
used  in  deducing  the  equa- 
tion of  the  epicycloid  we 
have  (see  Fig.  31), 


OC=a-b, 


and 


^  I 
X  =  ji^^ 


The  inclination  of  CP  to  the  negative  direction  of  the  axis  of 
X  is 

/       a  —  b. 


hence  the  equations  of  the  hypocycloid  are 

X  =  {a  —  b)  cos  f  +  b  cos  — ^ —  ip 


y  —  {a  —  b)  simp  —  b  sin  — ^ —  ^ 


(I) 


§  XXIL]  THE  FOUR-CUSPED  HYPOCYCLOID. 


l6l 


The  Four-Cusped  Hypocycloid. 


Fig.  32. 


161.  In  the  case  of  the 
hypocycloid  when  b  =  \a,  the 
circumference  of  the  rolling 
circle  is  one-fourth  the  circum- 
ference of  the  fixed  circle,  and 
the  curve  will  have  a  cusp  at 
each  of  the  four  points  where 
the  coordinate  axes  cut  the 
fixed  circle,  as  represented  in 
Fig.  32. 

On    substituting   \a    for  b 
equations  (2)  Art.  160  become 


x—\a  cos  0  +  \a  cos  3^ 
J  =  f^  sin  ^  —  \a  sin  3?/? 


(0 


Substituting  the  values  of  co^yp  and  sin  3^  from  the  for- 
mulas, 

cos  yp  —  ^  cos'  //'  —  3  cos  //', 


and 


sin  3^  =  3  sin  ?/'  —  4sin'*^;, 


we  have 


X  =  a 

y 


a  cos'  ^)  _ 
a  sin'  ^)  ' 


(2) 


whence        ;r^  —  <3:^  cos^  ^,         and        y^^  —  c^  ^\x^^. 

Adding,  we  have  x^-\-y^—a^, (3) 

the  rectangular  equation  of  the  curve. 


y 


CHAPTER    IX. 

Applications  of  the  Differential  Calculus  to 
Plane  Curves. 


XXIII. 

The  Equation  of  the   Tangent, 

(62.  The  equation  of  the  curve  being  given  in  the  form 
y  z=zf(x)^  the  inclination  of  the  tangent  at  any  point  is  deter- 
mined by  the  equation 

Hence,  if  (;ir,,  y^  be  a  point  of  the  curve,  the  equation  of  a 
tangent  at  (;i:„  y^  will  be  found  by  giving  to  the  direction- 
ratio  m^  in  the  general  equation 

y  —yx  =  m{x  —  X,), 

dy 


the  value     . 


;  thus 

or  y-y^=f\x^){x-X^) (2) 

For  example,  in  the  case  of  the  semi-cubical  parabola 
f^ax\ 


§  XXIII.]        THE  EQUATION  OF   THE    TANGENT.  163 

dy         „   3  /^ 

we  have  -t~  —  \\/  - • 

ax  ^  X 

The  point  {a,  a)  is  a  point  of  this  curve ;  the  equation  of 
the  tangent  at  this  point  is,  therefore, 

•^y  —  2x  =  a. 

The  Equation  of  the  NormaL 

163.  A  perpendicular  to  the  tangent  at  its  point  of  contact 
is  called  a  normal  to  the  curve. 

The  coordinate  axes  being  rectangular,  the  direction-ratio 
of  the  normal  is  the  negative  reciprocal  of  that  of  the  tangent ; 
for  the  inclination  of  the  normal  is  \n  +  ^,  and 

tan(J;r  +  ^)  =  _  cot  ^. 

The  equation  of  the  normal  may,  therefore,  be  written  thus — 

As  an  illustration,  let  us  take  the  equation  of  the  ellipse 

x'      f 

-  dy  h^x 

whence  —-— r-  . 

dx  ay 

The  equation  of  the  normal  at  any  point  {x^,  y^  of  the  ellipse 
is,  therefore. 


:64 


APPLICATIONS    TO  PLANE    CURVES.        [Art.    1 64. 


Subtangents  and  Subnormals, 


(64.  Denoting  by  s  the  length  of  the  arc  measured  from  some 

ds 
fixed  point,  —  denotes  the  velocity  of  P^  the  generating  point 

ttZ 

of  the  curve  ;  let  PT^  equal  to  ds,  be  measured  on  the  tangent 
at  P^  then  PQ  and  ^  7"  will  represent  dx  and  dy^  and  the  angle 
TPQ  will  be  (<> ;  hence  t 

dx  ^^ 


cos  ^  = 


ds 


sm 


and 


^j:  =   4/(^^t''  +  df). 


(2) 


Fig.  33. 


165.  The  distance  PT  (Fig.  34)  on  the  tangent  line  inter- 
cepted between  the  point  .of  contact 
and  the  axis  of  x  is  sometimes  called 
the  tangent,  and  in  like  manner  the  in- 
tercept PN  is  called  the  normal. 

From  the  triangles  PTR  and  NPR, 
we  have 
Fig.  34. 

/^r^j/cosec^=^J-=;.|/[i  +(^)], 


PN  —  y  sec  ^  —  y  -j-  —  y  y 


dx 


I  + 


The  projections  of  these  lines  on  the  axis  of  x,  that  is  TR 
and  RN,  are  called  the  sub  tangent  and  the  siibnormaL 
From  the  same  triangles,  we  have 


§  XXI 1 1.]         SUBTANGENTS  AND   SUBNORMALS.  1 65 

dx 
the  subtangent,  TR  =  y  cot  <f)—y—^ 

and  the  subnormal,    RN=  y  tan  (p  —  y  ~. 

ax 


The  Perpendicular  from  the  Origiit  upon  the  Tangent, 

\^^.  If  a  perpendicular/  to  the  tangent  Pi?  be  drawn  from 
the  origin,  we  have,  from  the   triangles  in  Fig. 

P/         35, 

/  —  ,1- sin  (;5  —  J/ cos  <;5,       .     .     .     (i) 


-^    ^  —  90°  being  taken  as  the  positive  direction  o/p. 
Substituting    the    values    of   sin  (f)   and    cos  (f>, 
Fig.  35.  equation  (i)  becomes 

_xdy  —  ydx  _   xdy  —  ydx  -    . 

^  ~         ds  ~  V{dx'-{-dy^) '^^ 

For  example,  let  us  determine  /  in   the  case   of  the  four- 
cusped  hypocycloid, 

X—  a  cos*  //?,  yz=z  a  sin'  7/7. 

Differentiating, 

dx  =  —  la  cos-  ^  sin  ?/?  dip,     and     dy  —  ^a  sin'' ^  costpdrp  ; 
'ivhence  ds  =  ^a  sin  ip  cos  ^'  ^y^?. 

Substituting  in  equation  (2)  we  obtain 
p  =  a  cos'  Jp  simp  -{-  a  sin'  ip  cos  ?/?  =  ^  sin  tp  cos  ?/?  =  ^(^;rj). 
To  ascertain  the  direction  of  /  it  is  necessary  to  determine 


l66  APPLICATIONS    TO   PLANE    CURVES.        [Art.    1 66. 

^.  The  ambiguity  in  the  value  of  (f>  as  determined  from  the 
equation  tan  ^  =  -^  may  be  removed  by  means  of  one  of  the 
formula^  of  Art.  164.     Thus,  in  the  present  case,  we  have 

tan  ^  =  —  tan  ^,    whence  ^  =  —  ^,     or     ^=:  n  ^  ip\ 

but,  since  cos  ^  =  — -  =  —  cos  ^, 

as 

we  must  take  <j)  =  7t  —  tp. 

The  direction  of/  when  positive  is  therefore  ^rt  —  ip. 

Examples  XXIII. 

I.  In  the  case  of  the  parabola  of  the  nth.  degree 

find  the  equations  of  the  tangent  and  the  normal  at  the  point  («,  a). 
y      2.  Find  the  subtangent  and  the  subnormal  of  the  parabola 

V     3.  Prove  that  the  subtangent  of  the  exponential  curve 

.    r       -J^ 

is   constant,  and  find   the   ordinate  of  the  point  of  contact  when  the 
tangent  passes  through  the  origin.  e. 

•\/     4.  Find  the  subnormal  of  the  ellipse  whose  equation  is 


a^       b' 

V 

5. 

Find  the 

subtangent 

of  the  curve 
an-^y  —  ^». 

§  XXIII.]  EXAMPLES.  167 

V       6.  In  the  case  of  the  parabola 

find/  in  terms  of  x. 


y^  =  4aXf 


For  the  upper  branch,  /  = 


V{a-\-x) 

7.  Find,  in  terms  of  ^,  the  equation  of  the  tangent  to  the  four- 
cusped  hypocycloid  (Art.  161),  and  thence  show  that  the  part  inter- 
cepted between  the  axes  is  of  constant  length. 

8.  In  the  case  of  the  epicycloid,  find  the  value  of  ds  in  terms  of 
the  auxiliary  angle  ^\     See  Art.  159. 

ds  —  2\a  -\-  b)  sm  —rdih. 
20 

9.  Determine  the  value  of  /  in  the  case  of  the  epicycloid  em- 
ploying the  value  of  ds  determined  in  the  preceding  example. 

/>  =  (^  +  2^)  sm  ^  . 


XXIV. 

Polar  Coordinates. 


167.  When  the  equation  of  a  curve  is  given  in  polar  co- 
ordinates the  vectorial  angle  6  is  usually  taken  as  the  inde- 
pendent variable ;  hence,  denoting  by  s  an  arc  of  the  curve,  it 
is  usual  to  assume  that  ds  and  dd  have  the  same  sign  ;  that  is, 

,        ds  .  .  . 

that  -j^  is  positive. 
du 

In  Fig.  36  let  PT,  a  portion  of  the  tangent  line,  represent 

ds ;  then,  producing  /-,  let  the  rectangle  PT  be  completed,  and 


1 68  APPLICATIONS    TO   PLANE   CURVES.        [Art.  1 67. 

let  ^  denote  the  angle  TPS\  that  Is,  the  angle  between  the 
positive  directions  of  r  and  s.  The  re- 
solved velocities  of  P  along  and  perpen- 

dr 
s    dicular  to  the    radius  vector  are  —  and 

-TT-,  the  latter  being  the  velocity  which  P 

would  have  if  r  were  constant  ;  that  is,  if 
P  moved  in  a  circle  described  with  ?-  as  a 
radius.     Hence  we  have 

PS  =  dr        and        PR  =  rdd. 

From  the  triangle  PST,  we  derive 

.       ,       rdS  .     ,       rdO  ^       dr  ,  , 

tan^.  =  ^,         smy.  =  --,         cosy.  =  -^^,    .     .    (i) 


Fig.  36. 


J  </5        ,  /r  o       fdr 


4-©'] « 


168,  The  second  of  equations  (i)  shows  that,  in  accordance 
with  the  assumption  that  ds  has  the  sign  of  dB,  the  value  of  ^ 
will  always  be  either  in  the  first  or  in  the  second  quadrant. 

The  first  of  equations  (i)  is  equivalent  to 

dr  ,  .- 

^°"/^  =  ^^' •  •  (3) 

which  shows  that  cot  ?/;  is  the  logarithmic  derivative  of  r  re- 
garded  as  a  function  of  Q.  Thus  in  the  case  of  the  logarithmic 
spiral 

r  =  at^^ 

we  have  log  r  =  log  a  +  nd, 

hence  cot  tp  =  « 


§  XXIV.] 


POLAR   COORDINATES. 


169 


whence  it  follows  that,  in  the  case  of  this  curve,  ip  is  constant. 
See  Art.  154. 

169.  It  is  frequently  convenient  to  employ  in  place  of  the 
radius  vector  its  reciprocal,  which  is  usually  denoted  by  u  ; 
then 


I  ,       ,  du 

r=-,     and     dr  —  — —^, 
u  u" 


(4) 


Making  these  substitutions  in  equations  (2)  and  (3)  we  have 

du 


ds  _  I 
de~u' 


and 


cot  ip  —  — 


ude 


(6) 


Polar  Subtangents  and  Subnormals, 


170,  Let  a  straight  line  perpendicular  to 
the  radius  vector  be  drawn  through  the  pole, 
and  let  the  tangent  and  the  normal  meet 
this  line  in  T  and  N  respectively ;  then  the 
projections  of  PT  and  PN  upon  this  line, 
that  '\^  OT  and  ON^  are  called  respectively 
the  po/ar  siibtangent  and  t\\Q  polar  subnormal. 
In  Fig.  37,  OPT=  tp  ;  whence 


OT=  r  tan  ib  =  r^  —  = , 

dr  du 


and 


Fig.  37  shows  that  the  value  of  OT  is  positive  when  its 


I70  APPLICATIONS   TO  PLANE   CURVES.        [Art.    170. 

direction  is  ^  —  90°  ;  that  of  ON  is,  on  the  other  hand,  positive 
when  its  direction  is  ^  +  90°. 


The  Perpendicular  from  the  Pole  upon  the  Tangent, 

171.  Let/  denote  the  perpendicular  distance  from  the  pole 
to  the  tangent ;  then,  from  Fig.  37  we  obtain 

These   expressions   give   positive   values   for  /,  because   -^  is 

assumed  to  be  positive,  and  Fig.  37  shows  that/  has  the  direc- 
tion (j)  —  90°,  ^  being  the  angle  which  the  positive  direction  of 
s  makes  with  the  initial  line. 

The  relation  between  /  and  u  is  obtained  thus :—  from  (i) 
we  have 

/  ~  r'dB'  ' 

and,  transforming  by  the  formulas  of  Art.  169, 

I  „       fdiiy' 


,ej (^> 

172.   The    expression    deduced    below    for    the    function 

+  im  is  frequently  useful. 
dtf 

Differentiating  (2),  we  have 

,  du  d^'u  2dp 


§  XXIV.]    THE  PERPENDICULAR   UPON  THE   TANGENT.       I /I 

hence  («4-^)^«=^|, 

,  dr 

or  since  du—  —  —^  y 


d'^'u  _T^    dp 
^  ^  'd&'~f"dr' 

The  Perpendicular  upon  an  Asymptote, 

173.  When  the  point  of  contact  P  passes  to  infinity  the 
tangent  at  P  becomes  an  asymptote,  and  the  subtangent 
OT  coincides  with  the  perpendicular  upon  the  asymptote. 
Hence  {Q^  denoting  a  value  of  Q  for  which  r  is  infinite)  the  length 

of  this  perpendicular  is  given  by  the  expression  —  -z-       ,  and 

like  the  polar  subtangent  is,  when  positive,  to  be  laid  off  in  the 
direction  0,  —  90°. 

This  expression  for  the  perpendicular  upon  the  asymptote 
is  also  easily  derived  by  evaluating  that  given  in  Art.  146. 
Thus — 

Points  of  Inflexion, 

174.  When,  as  in  Fig.  37,  the  curve  lies  between  the  tan- 
gent and  the  pole,  it  is  obvious  that  r  and  /  will  increase  and 

decrease  together ;  that  is,  -~  will  be  positive.     When  on  the 

dr 

other  hand  the  curve  lies  on  the  other  side  of  the  tangent, 
~~-  is  negative.  Hence  at  a  point  of  inflexion  -~  must  change 
sign. 


1/2  APPLICATIONS    TO  PLANE   CURVES.        [Art.   1 74. 

Now,  since/  is  always  positive,  it  follows  from  the  equation 
deduced  in  Art.  172  that  the  sign  of  this  expression  is  the  same 
as  that  of 

hence  at  a  point  of  inflexion  this  expression  must  change  sign, 

Mb.  As  an  illustration,  let  us  determine  the  point  of  in- 
flexion of  the  curve  traced  in  Art.  147 ;  viz., 

ae" 


e'-i 


In  this  case  u  =  -(i  —  6  "")  ; 

therefore  .  ^  J  =  i  (,  _,- _6.-) 

6'  -e^-e 


ad' 
Putting  this  expression  equal  to  zero,  the  real  roots  are 


and  it  is  evident  that,  as  6  passes  through  either  of  these  values, 

■-  .  d'^u   . 

the  expression  u  +  -^  cha 

flexion  are  determined  by 


d^u 
the  expression  u  +  -j^  changes  sign.     Hence  the  points  of  in 


e=  ±  ^2         and         r=^~. 
^  2 


§  XXIV.]  EXAMPLES.  173 

Examples   XXIV. 

1.  Prove  that,  in  the  case  of  the  lemniscata  r^  =  a*  cos  20, 

^  =  2e  +  irt,     and    |^  =  1 

2.  Find  the  subtangent  of  the  lituus  r"  =  —  ,  and  prove  that  the 
perpendicular  from  the  origin  upon  the  tangent  is 

3.  Find  the  polar  subtangent  of  the  spiral  r  {t^  +  e-^)  =  a. 

a 


0    c-tf 


e"  —  e 

4.  Find  the  value  of/  in  the  case  of  the  curve  r«  =  a""  sin  nO. 

I 
J>  =  a  (sin  «0) '     «"  • 

5.  In  the  case  of  the  parabola  referred  to  the  focus 

r  = ,  prove  that  /'  =  ar, 

I  +  cos  0  '  ^  ^ 

6.  In  the  case  of  the  equilateral  hyperbola 

r*  cos  20  =  a'f  prove  that/  =  — . 

7.  In  the  case  of  the  lemniscata 

r^  =  a^  cos  20,  prove  that/  =  —7 • 

CL 

8.  In  the  case  of  the  elHpse  r  =  — —  ,  the  pole  being  at 

^  1  —  e  cos  0 


174  APPLICATIONS   TO  PLANE   CURVES,  [Ex.  XXIV. 

the  focus,  determine/. 

9.  In  the  case  of  the  cardioid 

r  =  «  (i  +  cos  6),  prove  that  r^  =  2ap*. 

10.  Show  that  the  curve  r6  sin  0  =  «  has  a  point  of  inflexion  at 
which  r  =  — . 

7t 


XXV. 

Curvature. 


176.  If,  while  a  point  P  moves  along  a  given  curve  at  the 

ds 
rate  -^,  it  be  regarded  as  carrying  with  it  the  tangent  and 

normal  lines,  each  of  these  lines  will  rotate  about  the  moving 

point  P  at  the  angular  rate  -— - ,  ^  denoting  the  inclination  of 

the  tangent  line  to  the  axis  of  x. 

The  point  P  is  always  moving  in  a  direction  perpendicular 

to  the   normal   with  the  velocity  — ^.      Let    us   consider   the 

at 

motion  of  a  point  A  on  the  normal  at  a  given  dis- 
tance k  from  P  on  the  concave  side  of  the  arc. 
While  this  point  is  carried  forward  by  the  motion 

ds 
X  of  P  with  the   velocity  -7;  in  a  direction  perpen- 

^^'  ^  '       dicular  to  the  normal,  it  is  at  the  same  time  car- 
ried backward,  by  the  rotation  of  this  line  about  P,  with  the 


§  XXV.]  CUR  VA  TURK.  I J  5 

velocity  --3—  ;  since  this  is  the  velocity  with  which  A  would 

move  if  the  point  P  occupied  a  fixed  position  in  the  plane ; 
and  the  direction  of  this  motion  is  evidently  directly  opposite 
to  that  of  P.     Hence  the  actual  velocity  of  A  will  be 

ds        J  d<j) 
'dt~      ~dt' 


in  a  direction  parallel  to  the  tangent  at  P, 

Let  p  denote  the  value  of  k  which  reduces  this  expression 
to  zero,  and  let  C  (Fig.  38)  be  the  corresponding  position  of  A  ; 
then, 

ds  d(i> 

ds 
whence  PC  =  p~  -yj (i) 

a<p  ^ 

Ml.  The  value  of  p  determined  by  this  equation  is,  in 
general,  variable  ;  for,  if  the  point  Pmove  along  the  curve  with 

a  given  linear  velocity  -p,  the  angular  velocity  -J-  will  gene- 

rally  be  variable.     If  however  we  suppose  the  angular  velocity 

—z-  to  become  constant,  at  the  instant  when  P  passes  a  given 

position  on  the  curve,  -ry,  the  value  of  p.  will  likewise  become 
do 

constant,  and  C  will  remain  stationary.     When  this  hypothesis 

is  made,  the  curvature  of  the  path  of  P  becomes  constant,  for 

P  describes  a  circle  whose  centre  is  C,  and  whose  radius  is  p. 

Hence  this  circle  is  called  the  circle  of  curvature  corresponding 

to  the  given  position  oi  P\    C  \s  accordingly  called  the  centre  of 

curvature,  and  p  is  called  the  radius  of  curvature. 


1/6  APPLICATIONS    TO   PLANE   CURVES.        [Art.    1 78. 


The  Direction  of  the  Radiics  of  Curvature, 

178.  If  in  Fig.  38  the  arrow  indicates  the  positive  direction 
of  s ;  the  case  represented  is  that  in  which  ^  and  s  increase  to- 
gether, and  therefore  the  value  of  p  as  determined'by  equation 
(i),  Art.  176,  is  positive.  Hence  it  is  evident  that  when  p  is 
positive  its  direction  from  P  is  that  of  PC  in  Fig.  38  ;  namely, 
^  +  90°.  In  other  words,  to  a  person  looking  along  the  curve 
in  the  positive  direction  of  ds^  p,  when  positive,  is  laid  off  on  the 
left-hand  side  of  the  curve. 

For  example,  let  the  curve  be  the  four-cusped  hypocycloid, 

X  =  a  cos^  //;,  y  —  a  sin'  i[\ 

It  was  shown  in  Art.  166  that  for  this  curve 

ds  =  3^  sin  tp  cos  ?/?  dip,      and      ^  =  n  —  tp  \ 

hence  d(l)  —  —  dipj 

ds 
and  p  = -— =  —  3^:  sin  ?/?  cos  ^ (i) 

a(p 

When  y;  is  in  the  first  quadrant  p  is  negative ;  its  direction 
is  therefore  ^  -  :^7r  =  ^n  —  tp,  which  is  in  the  first  quadrant. 
When  ip  is  in  the  second  quadrant  p  is  positive  and  its  direction 
is  ^  +  ^71=  ^71  —  7pj  which  is  in  the  second  quadrant. 


The  Radius  of  Curvature  in  Rectangular  Coordinates. 

(79.  To  express  p  in  terms  of  derivatives  with  reference  to 
x^  we  have 


XXV.]  THE  RADIUS  OF  CURVATURE.  1 7/ 


hence 


'i  [-(£)? 


""I  ''-tr —y <■' 


dx  dx^ 


ds    . 


Since  -y-  is  assumed  to  be  positive,  (j)  should  be  so  taken  as 
dx 

to  cause  x  to  increase  with  s,  and  it  must  be  remembered  that 

the  direction  of  p  is  0  +  90°  when  p  is  positive,  in  accordance 

with  the  remark  in  the  preceding  article. 

180.  To  illustrate  the  application  of  the  above  formula,  we 
find  the  radius  of  curvature  of  the  ellipse 

y=±^^  V{a^-x^) (I) 

Differentiating,  :/-  =  =f  — 7-? — ^^y (2) 

dx  aV[a—x) 

.                                 dy                  ab  r  .. 

and  ^^qp— -r (s; 

dx        (a'  —  x-y 

Putting  b  —  aV(i  —  e")  we  obtain 

^  "^  \dx)  ~   a'-  x'   ' 
whence,  substituting  in  equation  (i)  of  the  preceding  article, 


178  APPLTCATIONS   TO  PLANE   CURVES.        [Art.    180. 

Expressions  for  9  in   which  x   ^i"  not   the  Independent 

Variable. 

181.  To  express  p  in  terms  of  derivatives  with  reference  to 
y^  we  have 


^';ir 


[■  *  (!)•] 


,              d(4>                df                                  L        V^j 
whence      ^  = ^„     and     p= ^ 

In  this  case  ds  and  ^  were  assumed  to  have  the  same  sign, 
hence  ^  must  be  taken  so  as  to  cause  j/  to  increase. 

182.  When  x  and  y  are  expressed  in  terms  of  a  third  vari- 
able we  employ  the  formula  deduced  below. 
Differentiating 

both  ^;ir  and  ^  being  regarded  as  variable,  we  have 

dx  dy  —  dy  d^x 

dx'  dxdy-dyd'x, 

^  ■*"  \dx) 

ds           {dx'  -h  dff^  (. 

whence  P  =  d^  =  dx  dy  ^  dyd'^x ^'^ 


XXV.]  EXAMPLES.  179 

Examples    XXV. 

1.  Find  the  radius  of  curvature  of  the  cycloid 

:x:  =  ^  (^  —  sin  ^'),  j;  nz  ^  (l  —  cOS  ^). 

ds 
Prove  that  #  =  J  (tt  —  ^),  and  use  p=  —; . 

p=  —  2  V{2ay). 

2.  Find  the  radius  of  curvature  of  \h.t. parabola  y^  =  /\ax. 

Va 


3.  Find  the  radius  of  curvature  of  the  catenary 


and  show  that  its  numerical  value  equals  that  of  the  normal  at  the 
same  point.     See  Art.  165. 


4.  Find  the  radius  of  curvature  of  the  semi-cubical  parabola 


af  =  x\ 


_  (4^?  +  ^x^x^ 


"=  6a 


5.  Find  the  radius  of  curvature  of  the  logarithmic  curve 

y  =  afr. 


cy 


l80  APPLICATIONS   TO  PLANE   CURVES.     [Ex.  XXV. 

6.  Find  the  radius  of  curvature  of  the  cissoid 


(2a  —  x)y 


_  a  Vx  (Sa  —  3^)^ 


7.  Find  the  radius  of  curvature  of  Xk^Q  parabola 

Vx  +  Vy  =  2  ^a. 


(^+# 


^  Va 


8.  Find  the  radius  of  curvature  of  the  cubical  parabola 


^  ~        6a'x 


9.  Find  the  radius  of  curvature  of  the  prolale  cycloid 
x  =  aip  —  bsinipj  y  =  a  —  b  cos  ^\ 


_  {a^  +  b"^  —  2ab  cos  tjS)^ 
'  b(a  cos  ^  —  b) 


XXVI. 
Envelopes, 


183.  The  curves  determined  by  an  equation  involving  x 
and  y  together  with  constants  to  which  arbitrary  values  may 
be  assigned  are  said  to  constitute  a  system  of  curves.  The 
arbitrary  constants  are  called  parameters.     When  but  one  of 


g  XXVI.]  ENVELOPES.  I8l 

the  parameters  is  regarded  as  variable,  denoting  it  by  a^  the 
general  equation  of  the  system  of  curves  may  be  expressed  thus  : 

f{x,y,a)=o (i) 

When  the  curves  of  a  system  mutually  intersect  (the  intersec- 
tions not  being  fixed  points),  there  usually  exists  a  curve 
which  touches  each  curve  of  the  system  obtained  by  causing  the 
value  of  a  to  vary. 

For  example,  the  ellipses  whose  axes  are  fixed  in  position, 
and  whose  semi-axes  have  a  constant  sum,  constitute  such  a 
system  ;  and,  if  we  regard  the  ellipse  as  varying  continuously 
from  the  position  in  which  one  semi-axis  is  zero  to  that  in  which 
the  other  is  zero,  it  is  evident  that  the  boundary  of  that  por- 
tion of  the  plane  which  is  swept  over  by  the  perimeter  of  the 
varying  ellipse  is  a  curve  to  which  the  ellipse  is  tangent  in  all 
its  positions.  A  curve  having  this  relation  to  a  given  system 
of  curves  is  called  the  envelope  of  the  system. 

Every  point  on  an  envelope  may  be  regarded  as  the  limit- 
ing position  of  the  point  of  intersection  of  two  members  of  the 
given  system  of  curves,  when  the  difference  between  the  cor- 
responding values  of  a  is  indefinitely  diminished.  For  this 
reason,  the  envelope  is  sometimes  called  the  locus  of  the  ultimate 
intersections  of  the  curves  of  the  given  system. 

184.  If  we  differentiate  equation  (i)  of  the  preceding  arti- 
cle (regarding  a:  as  a  variable  as  well  as  x  and  y)  the  resulting 
equation  will  be  of  the  general  form 

/i  (-^^  J>  oc)  dx  +  fj^x,  y,  a)  dy  -}-  fl{x,  y,  a)  da  =  o.   .     (2) 

In  this  equation  each  term  may  be  separately  obtained  by 
differentiating  the  given  equation  on  the  supposition  that  the 
quantity  indicated  by  the  subscript  is  alone  variable.  See 
Art.  64, 


1 82  APPLICATIONS   TO  PLANE   CURVES.        [Art.    1 84. 

From  equation  (2)  we  derive 

dy  ^  _  /j  {x,  y,  a)       f^  {x,  f,  a)  da 

dx  f'y  {x,  y,  a)       f'^  {x,  y,  a)  dx  *     *      *      *      ^3; 

In  Fig.  39  let  PC  be  the  curve  corresponding  to  a  particular 
value  of  or,  and  let  P  be  the  point  {x,  y)  ;  then  ^ 

the  expression  for  -j-    given  in  equation   (3) 

determines  the  direction  in  which  the  point  P 
is   actually   moving   when   x^  y,  and    a   vary   ^ 
simultaneously.  This  direction  depends  there-  Fig.  39. 

fore   in  part   upon   the   arbitrary  value  given  to  the  ratio  -7-  . 

185.  Now  if  a  were  constant  da  would   vanish,  and   equa- 
tion (3)  would  become 

^  =  _  /^(^>-^>^)  (.\ 

dx  f;{x,y,ay     ••••••     W 

This  expression  for  -~  determines   the  direction  in   which  P 

moves  when  PC  is  a  fixed  curve. 

Let  AB  be  an  arc  of  the  envelope,  and  let  C  be  its  point  of 
contact  with  PC,  Now,  if  P  be  placed  at  the  point  C,  it  is 
obvious  that  it  can  move  only  in  the  direction  of  the  common 
tangent  at  C^  whether  a  be  fixed  or  variable.  It  follows  there- 
fore that,  at   every  point   at  which  a  curve  belonging   to  the 

system  touches  the  envelope,  the  expressions  for  -—  given   in 

equations  (3)  and  (4)  must  be  identical  in  value. 

Assuming  that  f'^  (x,  y,  a)  and  /'  (x^  y,  a)  do  not  become  in- 
finite for  any  finite  values  of  x  andjj/,  the  above  condition  re- 
quires that 

/;(-^;/;«')  =  0 (5) 


§  XXVI.]  ENVELOPES.  183 

Hence  the  coordinates  of  every  point  of  the  envelope  must 
satisfy  simultaneously  equations  (5)  and  (i)  ;  the  equation  of 
the  envelope  is  therefore  obtained  by  eliminating  a  between 
these  two  equations. 

186.  Let  it  be  required  to  find  the  envelope  of  the  circles 
having  for  diameters  the  double  ordinates  of  the  parabola 

If  we  denote  by  a  the  abscissa  of  the  centre  of  the  variable 
circle,  its  radius  will  be  the  ordinate  of  the  point  on  the  para- 
bola of  which  a  is  the  abscissa,  the  equation  of  the  circle  will 
therefore  be 

y"^  +  {x  —  a)-  —  ^aa  —  O (l) 

Differentiating  with  reference  to  the  variable  parameter  «',  we 

have 

—  2  (,r  —  n')  —  4^  =  o, 

or  a  =  2a  -]r  X  \ (2) 

substituting  in  (i),  and  reducing,  we  obtain 

/^4a{a  +  x).    . (3) 

The  envelope  is,  therefore,  a  parabola  equal  to  the  given  para- 
bola and  having  its  focus  at  the  vertex  of  the  given  parabola. 

Two  Variable  Parameters. 

i87.  When  the  equation  of  the  given  curve  contains  two 
variable  parameters  connected  by  an  equation,  only  one  of 
these  parameters  can  be  regarded  as  arbitrary,  since,  by  means 
of  the  equation  connecting  them,  one  of  the  parameters  can 
be  eliminated.  Instead,  however,  of  eliminating  one  of  the 
parameters  at  once,  it  is  often  better  to  proceed  as  in  the  fol- 
lowing example. 


1 84  APPLICATIONS   TO  PLANE   CURVES.         [Art.  1 8/. 

Required,  the  envelope  of  a  straight  line  of  fixed  length  a, 
which  moves  with  its  extremities  on  two  rectangular  axes. 

Denoting  the  intercepts  on  the  axes  by  a  and  yS,  the  equa- 
tion of  the  line  is 

a  and  fi  being  two  variable  parameters  which,  by  the  condi- 
tions of  the  problem,  are  connected  by  the  relation 

a''-^  §^  =  c?.      . (2) 

Differentiating  (i)  and  (2)  with  respect  to  a  and  §  as  vari- 
ables, we  have 

xda      ydfi 
o?         fi' 


2-4-^  =  0, (3) 


and  ada  +  jSd^  =  o .     .     (4) 

We  have  now  four  equations  from  which  we  are  to  eliminate 

dcx. 
a^  y5,  and  the  ratio  — .     Transposing  and  dividing  (3)  by  (4), 
a  p 

we  obtain 

X  _  y 

Substituting  in  (i)  the  value  of  y  derived  from  the  I'ast 
equation,  we  have 

whence  by  equation  (2) 

a  =  x^a^. 


§  XXVI.]  EVOLUTES.  185 

In  like  manner  we  find 
Hence,  substituting  in  (2)  —  - 

The  envelope  is  therefore  a  four-cusped  hypocycloid. 

Evolutes, 


188.  In  Fig.  40  let  C  be  the  centre  of  curvature  of  the  given 
curve :  this  point  is  so  determined  (see  Art.  176)  as  to  have  no 
motion  in  a  direction  perpendicular  to  the  normal 

PC^  but  since  p  is  in  general  variable,  it  has  a  mo- 
tion in  the  direction  PC,     Hence  C  describes  a 
curve   to  which  the  normal  PC  is  always  tangent 
at  the  point  C.    Moreover,  since  P  has  no  motion 
in  the  direction  PC^  if  we  regard  P  as  a  fixed        p-j^    .^^ 
point  on  this  line,  the  rate  of  C  along  this  moving 
line  will  be  identical  with  its  rate  along  the  curve  which  it 
describes.     Hence  the  motion  of  PC  is  the  same  as  that  of  a 
tangent  line  rolling  upon  the  curve  described  by  (7,  while  P,  a 
fixed  point  of  this  tangent,  describes  the  original  curve. 

The  curve  described  by  the  centre  of  curvature  C  is  there- 
fore called  the  evolute  of  the  curve  described  by  P,  and  the 
latter  is  called  an  involute  of  the  former. 

189.  Since  the  evolute  of  a  given  curve  is  the  curve  to 
which  all  the  normals  to  the  given  curve  are  tangent,  it  is 
evidently  the  envelope  of  these  normals. 


1 86  APPLICATION'S   TO  PLANE   CURVES.         [Art.  1 89. 

The  equation  of  the  normal  at  the  point  (^,  y)  of  a  given 
curve  may  be  written  in  the  form 

x'-x  +  {y -y)£^=o, (i) 

(jr',y)  being  any  point  of  the  normal.     See  Art.  J63. 

In  this  equation  y  and  -^  are  functions  of  x  determined  by 

dX 

the  equation  of  the  given  curve,  and  x  is  to  be  regarded  as  the 
arbitrary  parameter.  Hence,  differentiating  with  reference  to 
;r,  we  have 

The  equation  of  the  evolute  is  therefore  the  relation  be- 
tween x'  and  y  which  arises  from  the  elimination  of  x  between 
equations  (i)  and  (2). 

190-  As  an  illustration,  let  it  be  required  to  find  the  evolute 
of  the  common  parabola 


y  =  2a^x^ ; 


dy  _  /a\^ 
dx~\^)' 


whence  -/=(-)   ,      and 


d^y 


dx       \x/  dx^  2x\ 

Substituting,  we  obtain  from  equation  (2)  of  the  preceding 
article 

x^=  —  ia^y ; 

whence,  from  equation  (i)  of  the  same  article, 

2yay^  =  4(x'  —  2df, 

the  equation  of  the  evolute,  which  is,  therefore,  a  semi-cubical 
parabola  having  its  cusp  at  the  point  (2^,  o). 


§  XXVI.]  EXAMPLES.  187 

191.  It  is  frequently  desirable  to  express  the  equation  of 
the  normal  in  terms  of  some  parameter  other  than  x  before 
differentiating.  Thus,  let  us  determine  the  evolute  of  the  ellipse 
by  means  of  the  equation  of  the  normal  in  terms  of  the  eccen- 
tric angle. 

The  equations  of  the  eUipse  are 

X  —  a  cos  ^',  and  jj/  =  ^  sin  ^ ; 

whence      dx=  —  a  sin  tp  dtp,        and         dy  —  b  cos  rp  dtp. 
Substitution  in  the  equation  of  the  normal, 

{x  —  x)dx  +  (y  —  y)  dy  —  o, 
gives       ax'  sin  ^  —  by'  cos  ^  —  (c^  —  Ij^)  sin  ^  cos  ^  =  o. 
Differentiating,  we  have 

ax'  cos  tp  +  by  sin  tp  —  {0?  —  b^)  (cos^  tp  —  sin^  tp)  z=  o; 
eliminating  y'  and  x^  successively,  and  dropping  the  accents, 

ax  =  {d^  —  IP)  cos^  tp         and         by  =  —  {c?  —  ^)  sin*  tp ; 
whence  {ax'f  +  {byf  =  (a^  -  IPf, 

Examples    XXVI. 

I.  Find  the  envelope  of  the  system  of  parabolas  represented  by  the 
equation 

y   ■=—{x-a\ 
in  which  a  is  an  arbitrary  parameter  and  c  a  fixed  constant. 


1 88  APPLICATIONS   TO  PLANE   CURVES.  [Ex.  XXVI. 

2.  Find  the  envelope  of  the  circles  described  on  the  double  or- 
dinates  of  an  ellipse  as  diameters. 

'2       I  7,2     ^' 


a"  +  ^'       b' 

3.  Find  the  envelope  of  the  ellipses,  the  product  of  whose  semi- 
axes  is  equal  to  the  constant  ^^ 

The  conjugate  hyperbolas,  2xy  =  ±  <^^ 

4.  Find  the  envelope  of  a  perpendicular  to  the  normal  to  the  para- 
bola, y  =  4aXy  drawn  through  the  intersection  of  the  normal  with  the 
axis. 

y  =  4a  {2a  —  x). 

5.  Find  the  envelope  of  the  ellipses  whose  axes  are  fixed  in  posi- 
tion, and  whose  semi-axes  have  a  constant  sum  c. 

The  f our-cusped  hypocycloid,  x^  +  y^  =  fi, 

6.  Given  the  equation  of  the  catenary 

prove  that 

/  ^  ^\ 

y'=  27,         and        x'=  x  —  —  Is"^  —  €    «j, 

and  deduce  the  equation  of  the  evolute. 

^'=  a  log  ->''  ±  (y"  -  4'^')^  ^  y  (y.  _  ^')i. 
2a  4a 

7.  Derive  the  equation  of  the  evolute  to  the  hyperbola,  its  equa- 
tions in  terms  of  an  auxiliary  angle  being 

X  =  asectp        and       y  =  b  tan  tp. 


§  XXVI.]  EXAMPLES.  189 

The  equation  of  the  normal  is 

ax  sin  ip  ^  by=  {(^  -\-  b^^  tan  ^, 

and  the  equation  of  the  evolute  is 

^!^l  —  b^y^  —  {a"  +  ^')3. 

8.  Find  the  equation  of  the  evolute  of  the  cycloid. 
The  equation  of  the  normal  is 

sin  if)  , 

X  -{■  y — 7  —  atp  —  o. 

I  —  cos  ip 

The  equations  of  the  evolute  are   . 

:x:  =  ^  (^  +  sin  ^)  and  y  =  —  a{i  —  cos^). 

The  evolute  is  therefore  a  cycloid  situated  below  the  axis  of  x, 
having  its  vertex  at  the  origin.     See  equations  (3),  Art.  158. 


CHAPTER    X. 

Functions  of  Two  or  More  Variables 


XXVII. 

The  Derivative  Regarded  as  the  Limit  of  a  Ratio, 

192.  The  difference  between  two  values  of  a  variable  is  fre- 
quently expressed  by  prefixing  the  symbol  A  to  the  symbol 
denoting  the  variable,  and  the  difference  between  correspond- 
ing values  of  any  function  of  the  variable,  by  prefixing  z/  to  the 
symbol  denoting  the  function.  Hence  x  and  x  +  Ax  denote 
two  values  of  the  independent  variable,  and  Af(x)  denotes  the 
difference  between  the  corresponding  values  oi  f{x)\  that  is, 

Ay  =  /f{_x)=f{x-^  Ax>)-f{x).     ...     (I) 
If  we  put  Ax  =  o,  we  shall  have  Ay  =  o; 

hence  the  ratio         ^_  ^f{x  ^  Ax)  - /{x) 

Ax  Ax  ^  ' 

takes  the  indeterminate  form  -  when  Ax  =  o.     The  value   as- 

o 

sumed  in  this  case  is  called  ^/le  limiting  value  of  the  ratio  of  the 
increments,  Ay  and  Ax,  when  the  absolute  values  of  these  incre- 
ments are  diminished  indefinitely. 

193.  To  determine  this  limiting  value,  for  a  particular  value 
a  of  X,  we  put  a  for  x  and  z  for  Ax  in  the  second  member  of 


§  XXVII.]    THE  DERIVATIVE  REGARDED  AS  A  LIMIT.  IQI 

equation  (2),  and  evaluate  for  ^  =  o,  by  the  ordinary  process 
(see  Art.  82).     Thus 

Z  Jo 


Therefore  when  Ax  is  diminished  indefinitely,  the  limiting  value 

of  —  corresponding  to  ;ir  =  ^ 

Ax 

value  of  Xj  we  have  in  general 


of  —  corresponding  to  ;ir  =  ^  is  -j-      ,  and,  since  <3:  denotes  any 
Ax  ax_\  a 


limit  of  -J-  —  -j~. 
Ax      ax 

A  V 
If  we  denote  by  e  the  difference  between  the  values  of  -~  and 
^  Ax 

—-.  we  shall  have 
ax 

%'%*■• w 

and  the  result  established  in  the  preceding  article  may  be  ex- 
pressed thus — 

^  =  o         when         Ax  —  6 ; 

in  other  words,  e  is  a  quantity  that  vanishes  with  Ax, 


Partial  Derivatives, 

194.  Let  t^=^fix,y\ 

in  which  x  and  y  are  two  independent  variables.     The  deriva- 
tive of  u  with   reference  to  x,  y  being  regarded  as  constant,  is 

denoted  by  ~-z-  u,  and  the  derivative  of  u  with  reference  to  r,  x 
ax 

being  constant,  by  ~j-  u.     These  derivatives  are  called  the  par" 
ay 

tial  derivatives  of  u  with  reference  to  x  and  y  respectively. 


192       FUNCTIONS  OF    TWO   OR  MORE    VARIABLES.     [Art.  1 94. 

Adopting  this  notation,  the  result  established  in  Art.  64  may 
be  expressed  thus ; 

du  =  — -  u  '  dx  H — —  U'  dy' 
dx  dy 

provided  u  denotes  a  function  that  can  be  expressed  by  means  of  the 
elemejttary  functions  differentiated  in  Chapters  II  and  III. 
It  is  now  to  be  proved  that  this  result  is  universally  true. 

195.  Let  AxU  denote  the  increment  of  ti  corresponding  to 
Ax^  y  being  unchanged,  AyU  the  increment  corresponding  to  Ay, 
X  being  unchanged,  and  Au  the  increment  which  u  receives 
when  x  and  y  receive  the  simultaneous  increments  Ax  and  Ay, 
Let 

u  ~f{x  +  Ax,y)y 

and  m"  =f{x  +  Ax,  y  +  Ay) ; 

then  A.xU  ^^  u  ~  u, 

AyU  =  u''  —  u\ 

and  Au  =  u"  —  u\ 

hence  Au  =  AxU  -k-  Ayu' (i) 

Denoting  by  At  the  interval  of  time  in  which  x,  y,  and  u  re- 
ceive the  increments  Ax,  Ay,  and  Au,  we  have 

Au  __  AxU      AyU'  ,  . 

Since  Au  Is  the  actual  increment  of  u  in  the  interval  At,  the 

du 
limit  of  the  first  member  of  equation  (2)  is,  by  Art.  ig^,  —  ,  the 

ai 

A  u 
rate  of  u.     The  limit  of  -^r-  is  the  rate  which  u  would  have 

At 


§  XXVII.]  PARTIAL   DERIVATIVES.  I93 

were  x  the  only  variable  ;  and,  since  -j-u-  dx  \s  the  value  which 
du  assumes  when  this  supposition  is  made,  if  we  put 

-r-u  •  dx  =  dj:U.  ~~  ~ 

dx 

d  u 
this  rate  will  be  denoted  by  --J— .     Hence  by  equation  (4),  Art. 

193,  equation  (2)  becomes 

du    ,  dj^u  .     ,     dyu'        „ 

in  which  e,  e\  and  e"  vanish  with  At ;  but  when  At  =  o,  Ax  =  o, 
and  therefore  u'  =  u  ;  hence,  putting  At  =  o,  we  have 


du 

dt 

dxU      dyU 

"  dt   ^   dt' 

du 

=:d^u  -\-  dyU\ 

Therefore 

that  is,  du  —  -j-U'dx-\--r-U'  dy, 

dx  dy 

196.  This  result  is  usually  written  in  the  form 

,        du  J        du    - 
du=  -r-dx  ^-  -j-dy, 
dx  dy 

but  when  written  in  this  form  it  must  be  remembered  that  the 
fractions  in  the  second  member  represent  partial  derivatives^ 
the  symbol  du  in  the  numerators  standing  for  the  quantities 
denoted  above  \yy  d^cU  and  dyU,  which  are  sometimes  Q.d\\^  A  par- 
tial differentials.  The  du  that  appears  in  the  first  member  is 
called  the  total  differential  of  u  when  x  and  y  are  both  variable. 
The  above  result  is  easily  extended  to  functions  of  more 
than  two  independent  variables. 


\(^^  FUNCTIONS   OF  TWO   OR  MORE    VARIABLES.  [Ex.  XXVII. 

Examples   XXVII. 

1.  Given      u  —  (x^  +  J^*)^,  prove  that 

du         du       _ 

XV 

2.  Given      u  =  — =^-—  ,  prove  that  v 

X  -\- y 

du  du  _ 

dx         dy         ' 


3.  Given      w  =  tan"'  f — l~'i'>  P^ove  that 


du    ,       du 

x-j-  +y-j-  =  0. 
dx  dy 


4.  Given       u  =  logyX^  to  find         md   — . 


^  __       I      ^      du  _^   ^  logjp 
^/^e      ^  log j;  *      dy~~  y  (log^')^ 


XXVIII. 

The  Second  and  Higher  Derivatives  regarded  as  Limits. 
197.  In  Art.  193  it  is  shown  that 

Jy  _  dy 
Ax       dx 

In  this  equation  ^  is  a  function  of  x  and  likewise  of  Ax-,  hence 

de 
the  derivative  -j-  is  in  general  a  function  of  x  and  of  Ax.    It  is 


§  XXVIIL]      THE    SECOND   DERIVATIVE   AS  A    LIMIT.         I95 

also  proved  in  the  same  article  that  e  becomes  zero  when  Ax 
vanishes;  that  is,  e  assu7ttes  a  constant  value  independent  of  the 
value  of  X  when  Ax  becomes  zero  ;  hence,  when  Ax  is  zero,  the 
derivative  of  e  with  reference  to  x  must  take  the  value  zero, 
whatever  be  the  value  of  ;r ;  in  other  words, 

—r-  vanishes  with  Ax, 
dx 

In  a  similar  manner  it  may  be  shown  that  each  of  the  higher 
derivatives  of  e  with  reference  to  x  vanishes  when  Ax  —  O. 

198.  Since  ~  is  a  function  of  ;r,  A  -f-  will  denote  the  incre- 
Ax  Ax 

ment   of  this  function  corresponding  to  Ax,     Employing  the 

symbol  ——  to  denote  the  operation  of  taking  this  increment, 

and  dividing  the  result  by  Ax,  we  obtain,  by  applying  to  this 
function  the  principle  expressed  in  equation  (4),  Art.  193, 

A     Ay        d     Ay        ,  ,  . 

Ax    Ax      dx    Ax  ^  ^ 


-m-'Y^ 


d'y      ... 


de^ 
dx^   '   dx 


de 
In  this  equation  both  e'  and  --j-  vanish  with  Ax  by  the  preced- 
ing article  ;  hence  the  sum  of  these  quantities  likewise  vanishes 
with  Ax^  and  may  be  denoted  by  e.     Thus  we  write 

^.^=J>  +  ,. (2) 

Ax    Ax      dx^  ^  ^ 


196        FUNCTIONS  OF   TWO   OR   MORE    VARIABLES,   [Art.  I99. 

199.  Since  Ax  is  an  arbitrary  quantity  it  may  be  regarded 

Ay 
as  constant,  whence  A  -~  is  the  increment  of  a  fraction  whose 
Ax 

denominator  is  constant ;  but  this  is  evidently  equivalent  to  the 

result  obtained  by  dividing  the  increment  of  the  numerator  by 

the  denominator ;  that  is, 

J  4r^  A'Ay  ^ 
Ax        Ax 

The  numerator  A  -  Ay  is  usually  denoted  by  the  symbol  A^y\ 
hence  equation  (2)  may  be  written  thus : 

-^  =  ^-^  +  .                              .  (3) 

Ax'       dx'^'' ^^^ 

and,  since  e  vanishes  with  Ax^  it  follows  that  the  second  deriva- 
tive is  the  limit  of  the  expression  in  the  first  member  of  equa- 
tion (3). 

In  a  similar  manner  it  may  be  shown  that  each  of  the  higher 
derivatives  is  the  limit  of  the  expression  obtained  by  substi- 
tuting A  for  d  in  the  symbol  denoting  the  derivative. 


Higher  Partial  Derivatives, 

200.  The  partial  derivatives  of  u  with  reference  to  x  and  y 
are  themselves  functions  of  x  and  y.  Their  partial  derivatives, 
viz., 

d    du       d     dti       d     du  a       ^     ^^^ 

dx   dx"*    dy    dx''    dx    dy^  dy    dy^ 

are  called  partial  derivatives  of  u  of  the  second  order. 

It  will  now  be  shown  that  the  second  and  third  of  these 
derivatives,  although  results  of  different  operations,  are  in  fact 
identical ;  that  is,  that 

d     du  _   d    du 
dy    dx     dx    dy* 


§  XXVI 1 1.]       HIGHER  PARTIAL   DERIVATIVES.  1 97 

Employing  the  notation  introduced  in  Art.  195,  we  have 

A^u=f{x  +  Ax,y)-  f{x,y)', 

if  in  this  equation  we  replace  y  hy  y  +  Ay,  we  obtain  a  new 
value  of  Aj^u,  and,  denoting  this  value  by  A'ji,  we  have 

A'ji  =f{x+  Ax,  y  +  Ay)  -  f  {x,  y  +  Ay). 

Since  this  change  in  the  value  of  A^u  results  from  the  increment 
received  by  y,  the  expression  for  the  increment  received  by 
A^u  will  be  Ay  (A^^u)  ;  hence 

Ay  {Aj,u)  =  A'^u  —  Aj,u, 
or 

Ay  {A^7c)=f{x  +  Ax,  y^Ay)-f{x,y  +  Ay)-f{x  +  Ax,  })-^f{x,y). 

The  value  of  A^^Ayu),  obtained  in  a  precisely  similar  manner, 
is  identical  with  that  just  given  ;  hence 

Ay{A,u)  =  A^{AyU) (I) 

Since  Ax  is  constant,  we  have,  as  in  Art.  199, 

^y  {^:cU)  ^   ^     ^    A^ 

Ax  ^  '  Ax' 

Hence,  dividing  both  members  of  equation  (i)  by  Ax  •  Ay,  we 
have 

Ay  ^  A^u  _  4r     AyU  ,  . 

Ay'  Ax       Ax'  Ay' ^  ' 

or,  employing  the  symbol  —  as  in  Art.  198, 

A      A  A      A 

Ay    Ax  Ax    Ay 


198      FUNCTIONS  OF   TWO   OR   MORE    VARIABLES.      [Art.  200. 


From  this  result,  by  a  course  of  reasoning  similar  to  that  em- 
ployed in  Art.  198,  we  obtain 

d    du        d     du  f  . 
—  =  —  .  — - (3) 

dy    dx       dx    dy 

201.  The  partial  derivatives  of  the  second  order- are  usually 
denoted  by 

^u  d\i  dHc^ 

dx'*  dxdy'  dy'' 

the  factors  dx  and  dy  in  the  denominator  of  the  second  being, 
by  virtue  of  formula  (3),  interchangeable,  as  in  the  case  of  an 
ordinary  product. 

The  numerators  of  the  above  fractions  are  of  course  not 
identical.     Compare  Art.  196. 

Formula  (3)  of  the  preceding  article  is  readily  verified  in 
any  particular  case.     Thus,  given 

u=.y^y 

.  du  .  .         du  ^  ^ 

whence         -r-  =  y^^og-y,        and        — -  =  xy"^'^ ; 
dx      ^  dy 

d     du  ,    ,  .        d    du 

-.-=^^-(.l0g^+l):=^.^. 

nxamples  XXVIII. 

1.  Given        u  =  sec  {y  +  ax)  +  tan  {y  —  ax),  prove  that 

^""^  d/' 

2.  Verify  the  theorem    ,     ,   =    ,    ,    when  u  =  sin  (jc/). 

^  dxdy       dyax 

3.  Verify  the  theorem       ^    =  -j^  ^^^^  ^  ~  ^^^  ^^^  ^^^  "^  •^'^' 


§  XXVIII.] 


EXAMPLES. 


199 


4.  Verify  the  theorem    ,  „   .    =  — — r^  when  u  =  tan      — . 

ay  ax      ax  ay  y 

5.  Verify  the  theorem  ^  =       ^      when  «^  =  jj/  log  (i  •{-  ^\ 


dy  dx^       dx^  dy 
6.  Given        2^  =  sinjc  cosj,  prove  that 

d^u  d*u  d*u 


df  dx^       dx^  dy^      dxdydxdy* 
7.  Given        u  =  x^z*  -\-  s'y'^z^  +  x^y^z^^  derive 
d'u 


dx^  dy  dz 


=  ^y^z""  +  2>yz. 


8.  Given 


i/(4^^-^) 


,  prove  that 


//'  d"" 


dadb 


9.  Given        «  =  (^  +  jj;)',  prove  that 


d^'^?/  ^/V   _  du 

dx^  dx  dy~  dx' 


10.  Given        u 


j7,  prove  that 


^^2^  ^2^         ^^  _ 

d^^lf^~d?~'^' 


AN 


ELEMENTARY  TREATISE 


INTEGRAL  CALCULUS 


FOUNDED   ON  THE 


xMETHOD  OF  RATES  OR  FLUXIONS 


WILLIAM   WOOLSEY   JOHNSON 

PROFESSOR  OF  MATHEMATICS   AT  THE    UNITED   STATES  NAVAL  ACADEMY 
ANNAPOLIS  MARYLAND 


FOURTH     THOUSAND. 


NEW   YORK: 

JOHN    WILEY    AND    SONS, 

53  East  Tenth  Street, 

1893. 


CoPVKtGHT, 

1881, 

By  JOHN  WILEY  AND  SONS. 


PHESS  OF   J.   J.   LITTLE  L  CO., 
NOS.    10   TO   to    A3T0R    PLACE,    NEW   YORK. 


PREFACE. 


This  work,  as  at  present  issued,  is  designed  as  a  shorter 
course  in  the  Integral  Calculus,  to  accompany  the  abridged 
edition  of  the  treatise  on  the  Differential  Calculus,  by  Pro- 
fessor J.  Minot  Rice  and  the  writer.  It  is  intended  hereafter 
to  publish  a  volume  commensurate  with  the  full  edition  of  the 
work  above  mentioned,  of  which  the  present  shall  form  a  part, 
but  which  shall  contain  a  fuller  treatment  of  many  of  the  sub- 
jects here  treated,  including  Definite  Integrals,  and  the  Me- 
chanical Applications  of  the  Calculus,  as  well  as  Elliptic  Inte- 
grals, Differential  Equations,  and  the  subjects  of  Probabilities 
and  Averages.  The  conception  of  Rates  has  been  employed 
as  the  foundation  of  the  definitions,  and  of  the  whole  subject 
of  the  integration  of  known  functions.  The  connection  be- 
tween integration,  as  thus  defined,  and  the  process  of  summa- 
tion, is  established  in  Section  VII.  Both  of  these  views  of  an 
integral — namely,  as  a  quantity  generated  at  a  given  rate,  and 
as  the  limit  of  a  sum — have  been  freely  used  in  expressing 
geometrical  and  physical  quantities  in  the  integral  fo^-fr. 


HI 


IV  PREFACE, 


The  treatises  of  Bertrand,  Frenet,  Gregory,  Todhunter,  and 

Williamson,  have  been  freely  consulted.     My  thanks  are  due 

to  Professor  Rice  for  very  many  valuable  suggestions  in  the 

course  of  the  work,  and  for  performing  much  the  larger  share 

:)f  the  work  of  revising  the  proof-sheets. 

-W.  W.  J. 

U.  S.  Naval  Academy,  July,  i88i. 


CONTENTS. 

CHAPTER  1. 

Elementary  Methods  of  Integration. 
I. 

PAGE 

Integrals - . .  I 

The  differential  of  a  curvilinear  area 3 

Definite  and  indefinite  integrals 4 

Elementary  theorems 6 

Fundamental  integrals. ...    7 

Examples  I lo 

II. 

Direct  integration ..,,..,, 14 

Rational  fractions 15 

Denominators  of  the  second  degree 16 

Denominators  of  degrees  higher  than  the  second 19 

Denominators  containing  equal  roots 22 

Examples  II 26 

III. 

Trigonometric  integrals  33 

Cases  in  which     sin"^  0  cos^  Q  de  is  directly  integrable 34 

The  integrals     sin*  e  dd,  and     cos"  ede 36 


The  integrals  f  -r-^"—  ,     f  ^  ,  and   f 
J  sm  0  cos  d      J  sm  0  J 


^, 37 

cosei 


V 


VI  CON-TENTS. 

PAGE 

Miscellaneous  trigonometric  integrals .38 

The  integration  of ; 40 

^  a  -\-  b  cose  ^ 

Examples  III ; 43 


CHAPTER  II. 

Methods  of  Integration — Continued. 

IV. 

Integration  by  change  of  independent  variable 50 

Transformation  of  trigonometric  forms 51 

Limits  of  a  transformed  integral 53 

The  reciprocal  of  x  employed  as  the  new  independent  variable 53 

A  power  of  x  employed  as  the  new  independent  variable 54 

Examples  IV 56 

V. 

Integrals  containing  radicals 59 

Radicals  of  the  form  .^{ax^  +  b) 61 

dx 
The  integration  of — 64 

\/{x'^  ±  a') 

Transformation  to  trigonometric  forms 65 

Radicals  of  the  form   \/(ax'^  +  bx  +  c) t 67 

dx                        ,                      dx 
^[{x  -  a){x  ~  fJ)]     ^"^      J  V[(^-a')(/^-^)] 
Examples  V 70 


The  integrals  I — and     f — 68 


VI. 

Integration  by  parts 77 

A  geometrical  illustration 78 

Applications 78 

Formulas  of  reduction 81 


Reduction  of    sin"'  e  de    and      ( 
tion  of  I  si] 


cos'«  ede 82 

Reduction  of  I  sin"^  e  cos«  edQ 84 


CONTENTS.  Vll 


PAGE 


Illustrative  examples 87 

Extension  of  the  formula  employed  in  integration  by  parts 8g 

Taylor's  theorem 90 

Examples  VI 91 

VII. 

Definite  integrals * 97 

Multiple-valued  integrals 100 

Formulas  of  reduction  for  definite  integrals loi 

Elementary  theorems  relating  to  definite  integrals    104 

Change  of  independent  variable  in  a  definite  integral 105 

The  differentiation  of  an  integral 106 

Integration  under  the  integral  sign 109 

The  definite  integral  regarded  as  the  limiting  falue  of  a  sum iii 

Additional  formulas  of  integration 115 

Examples  VII 117 


CHAPTER    III. 

Geometrical  Applications. 

VIII. 

Areas  generated  by  variable  lines  having  fixed  directions 123 

Application  to  the  witch 124 

Application  to  the  parabola  when  referred  to  oblique  coordinates 126 

The  employment  of  an  auxiliary  variable 126 

Areas  generated  by  rotating  variable  lines 12S 

The  area  of  the  lemniscata 129 

The  area  of  the  cissoid 130 

A  transformation  of  the  polar  formulas 130 

Application  to  the  folium 131 

Examples  VIII 134 

IX. 

The  volumes  of  solids  of  revolution 141 

The  volume  of  an  ellipsoid 143 

Solids  of  revolution  regarded  as  generated  by  cylindrical  surfaces   144 

Double  integration 145 

Determination  of  the  volume  of  a  solid  by  double  integration 149 


Vlll  CONTENTS. 


PAGE 

The  determination  of  volumes  by  triple  integration 150 

Elements  of  area  and  volume 152 

Polar  elements 154 

The  determination  of  volumes  by  polar  formulas 155 

Polar  coordinates  in  space 157 

Application  to  the  volume  generated  by  the  revolution  of  a  cardioid 159 

Exaj7iples  IX 160 


X. 

Rectification  of  plane  curves 168 

Rectification  of  the  semi-cubical  parabola 168 

Rectification  of  the  four-cusped  hypocycloid 169 

Change  of  sign  oi  ds 1 70 

Polar  coordinates ! 1 70 

Rectification  of  curves  of  double  curvature 171 

Rectification  of  the  loxodromic  curve ; 172 

Examples  X 173 


XI. 

Surfaces  of  solids  of  revolution 178 

Quadrature  of  surfaces  in  general 179 

The  expression  in  partial  derivatives  for  sec  v i&o 

The  determination  of  surfaces  by  polar  formulas 181 

Examples  XI 183 

XII. 

Areas  generated  by  straight  lines  moving  in  planes 186 

Applications 187 

Sign  of  the  generated  area 189 

Areas  generated  by  lines  whose  extremities  describe  closed  circuits 190 

Amsler's  Planimeter 191 

Examples  XII 193 


XIII. 

Approximate  expressions  for  areas  and  volumes 195 

Simpson's  rules 197 

Cotes'  method  of  approximation 19S 


CONTENTS.  IX 


PAGE 

Weddle's  rule 199 

The  five-eight  rule 199 

The  comparative  accuracy  of  Simpson's  first  and  second  rules 2CX) 

The  application  of  these  rules  to  solids .' ^ 200 

WooUey's  rule 201 

Examples  XIII 202 


CHAPTER    IV. 

Mechanical  Applications. 
XIV. 

Definitions 204 

Statical  moment 204 

Centres  of  gravity 206 

Polar  formulas 208 

Centre  of  gravity  of  the  lemniscata 209 

Solids  of  revolution 2og 

Centre  of  gravity  of  a  spherical  cap 210 

The  properties  of  Pappus 210 

Examples  XIV 212 

XV. 

Moments  of  inertia 219 

Moment  of  inertia  of  a  straight  line 220 

Radii  of  gyration 220 

Radius  of  gyration  of  a  sphere 221 

Radii  of  gyration  about  parallel  axes 222 

Application  to  the  cone 223 

Pola ;  moments  of  inertia 225 

Examples  XV 225 


THE 

INTEGRAL    CALCULUS 


CHAPTER    I. 
Elementary  Methods  of  Integration. 


I. 

Integrals, 

I.  In  an  important  class  of  problems,  the  required  quanti- 
ties are  magnitudes  generated  in  given  intervals  of  time  with 
rates  which  are  either  given  in  terms  of  the  time  /,  or  are 
readily  expressed  in  terms  of  the  assumed  rate  of  some  other 
independent  variable. 

For  example,  the  velocity  of  a  freely  falling  body  is  known 
to  be  expressed  by  the  equation 

v=gt,     .........     (i) 

in  which  t  is  the  number  of  seconds  which  have  elapsed  since 
the  instant  of  rest,  and  ^  is  a  constant  which  has  been  deter- 
mined experimentally.     If  s  denotes  the  distance  of  the  body 


2  ELEMENTARY  METHODS   OF  INTEGRATION.    [Art.  I. 

at  the  time  /,  from  a  fixed  origin  taken  on  the  line  of  motion, 
V  is  the  rate  of  s ;  that  is, 

ds 
''  =  di' 

hence  equation  (i)  is  equivalent  to 

ds  =  gtdt,  .  - .     '     (2) 

which  expresses  the  differential  of  s  in  terms  of  /  and  dt.  Now 
it  is  obvious  that  ^g^^  is  a  function^of  /  having  a  differential 
equal  to  the  value  of  ds  in  equation  (2) ;  and,  moreover,  since 
two  functions  which  have  the  same  differential  (and  hence  the 
same  rate)  can  differ  only  by  a  constant,  the  most  general 
expression  for  s  is 

s  =  iP'+C, .     (3) 

in  which  C  denotes  an  undetermined  constant. 

2.  A  variable  thus  determined  from  its  rate  or  differential 
is  called  an  integral,  and  is  denoted  by  prefixing  to  the  given 

differential  expression  the  symbol    ,  which  is  called  the  integral 

sign."'^     Thus,  from  equation  (2)  we  have 


-\^ 


dt. 


which  therefore  expresses  that  i-  is  a  variable  whose  differential 
IS  gtdt;  and  we  have  shown  that 

gtdt  =  ^gt^  +  C. 

The  constant  C  is  called  the  constant  of  integration ;  its 
occurrence  in  equation  (3)  is  explained  by  the  fact  that  we 
have  not  determined  the  origin  from  which  s  is  to  be  measured. 

*  The  origin  of  this  symbol,  which  is  a  modification  of  the  long  j,  will  be 
explained  hereafter.    See  Art.  100. 


§  I.]       THE  DIFFERENTIAL    OF  A    CURVILINEAR  AREA.  3 

If  we  take  this  origin  at  the  point  occupied  by  the  body  when 
at  rest,  we  shall  have  ^  =  o  when  /  =  o,  and  therefore  from 
equation  (3)  6^=0;  whence  the  eqtiation  becomes  s  —  \gt^. 


The  Differential  of  a  CurviUnear  Area, 

3.  The  area  included  between  a  curve,  whose  equation  is 
given,  the  axis  of  x  and  two  ordinates  affords  an  instance  of 
the  second  case  mentioned  in  the  first  paragraph  of  Art.  I  ; 
namely,  that  in  which  the  rate  of  the  generated  quantity,  al- 
though not  given  in  terms  of  /,  can  be  readily  expressed  by  means 
of  the  assumed  rate  of  some  other 
independent  variable. 

Let  BPD  in  Fig.  i  be  the  curve 
whose  equation  is  supposed  to  be 
given  in  the  form 

Supposing   the   variable    ordinate 

PR  to  move  from  the  position  AB 

to  the  position   CD^  the  required 

area  ABDCis  the  final  value  of  the  Fig.  i. 

variable  area  ABPR,  denoted  by 

^,  which  is  generated  by  the  motion  of  the  ordinate.     The  rate 

at  which  the  area  A  is  generated  can  be  expressed  in  terms  of 


the  rate  of  the  independent  variable  x. 


dA  dx 

assumed  rates  are  denoted,  respectively,  by  -—-  and  -— 

dt  dt 


The   required  and  the 
and,  to 


express  the  former  in  terms  of  the  latter,  it  is  necessary  to 
express  dA  in  terms  of  dx.  Since  x  is  an  independent  variable, 
we  may  assume  dx  to  be  constant ;  the  rate  at  which  A  is  gen- 
erated is  then  a  variable  rate,  because  PR  or  y  is  of  variable 
length,  while  moving  at  a  constant  rate  along  the  axis  of  x. 
Now  dA  is  the  increment  which  A  would  receive  in  the  time 


4  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  3. 

dt,  were  the  rate  of  A  to  become  constant  (see  Diff.  Calc, 
Art.  17).  If,  now,  at  the  instant  when  the  ordinate  passes  the 
position  PR  in  the  figure,  its  length  should  become  constant, 
the  rate  of  the  area  would  become  constant,  and  the  increment 
which  would  then  be  received  in  the  time  dt,  namely,  the 
rectangle  PQSRy  represents  dA.  Since  the  base  RS  of  this 
rectangle  is  dx,  we  have 

dA—ydx—f{x)dx (1) 

Hence,  by  the  definition  given  in  Art.  2,  A  is  an  integral,  and 
is  denoted  by 

A^\^f{x)dx .     (2) 


Definite  Integrals. 

4.  Equation  (2)  expresses  that  y^  is  a  function  of  x^  whose 
differential  \^f{x)dx  ;  this  function,  like  that  considered  in  Art. 
2,  involves  an  undetermined  constant.  In  fact,  the  expres- 
sion    f{x)dx  is  manifestly  insufficient  to  represent  precisely 

the  area  ABPR,  because  OA,  the  initial  value  of  x,  is  not  indi- 
cated. The  indefinite  character  of  this  expression  is  removed 
by  writing  this  value  as  a  subscript  to  the  integral  sign  ;  thus, 
denoting  the  initial  value  by  a^  we  write 


\^f{x)dx, (3) 


in  which  the  subscript  is  that  value  of  x  for  which  the  integral 
has  the  value  zero. 

If  we  denote  th.Q  final  value  of  x  (OC  in  the  figure)  by  d,  the 
area.  A BDC,  which   is  a  particular  value  of  Aj  is  denoted  by 


§  I.]  '  DEFINITE  INTEGRALS.  5 

writing   this    value    of  x    at    the    top    of   the   integral   sign, 
thus, 


ABDC 


\A^)dx (4) 


This  last  expression  is  called  a  definite  integral^  and  ^  and 
b  are  called  its  limits.      In  contradistinction,  the  expression 

f(x)dx  is  called  an  indefinite  integral, 

5.  As  an  application  of  the  general  expressions  given  in  the 
last  two  articles,  let  the  given  curve  be  the  parabola 

Equation  (2)  becomes  in  this  case 

A  ={:t^dx. 

Now,  since  ^x^  is  a  function  whose  differential  is  x^dx^  this 
equation  gives 

A={x'dx  =  j^x^-h  C, (I) 

in  which  C  is  undetermined. 

Now  let  us  suppose  the  limiting  ordinates  of  the  required 
area  to  be  those  corresponding  to  ;r  =  i  and  x  =  ;^.  The  vari- 
able area  of  which  we  require  a  special  value  is  now  represented 

by  [  x^dxy  which  denotes  that  value  of  the  indefinite  integral 

which  vanishes  when  x  =  1.     If  we  put  ;ir  =  i  in  the  general 
expression  in  equation  (i),  namely  ^x^  +  C,  we  have  ^  +  C;- 
hence  if  we  subtract  this  quantity  from  the  general  expression, 
we  shall  have  an  expression  which  becomes  zero  when  x  =  i. 
We  thus  obtain 

A={x^dx=ix'-i. 


6  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  5, 

Finally,  putting,  in  this  expression  for  the  variable  area,  x  =  3, 
we  have  for  the  required  area 


6.  It  is  evident  that  the  definite  integral  obtained  by  this 
process  is  simply  the  difference  between  the  values  of  the  indefinite 
integral  at  the  upper  and  lower  limits.  This  difference  may  be 
expressed  by  attaching  the  limits  to  the  symbol  ]  affixed  to  the 
value  of  the  indefinite  integral.  Thus  the  process  given  in  the 
preceding  article  is  written  thus. 


jydx  =  -i:^  +  cj=9-i  = 


The  essential  part  of  this  process  is  the  determination  of 
the  indefinite  integral  or  function  whose  differential  is  equal  to 
the  given  expression.  This  is  called  the  integration  of  the 
given  differential  expression. 

Elementary    Theorems, 

7.  A  constant  factor  may  be  transferred  from  one  side  of  the 
integral  sign  to  the  other.  In  other  words,  \{  m  is  a  constant 
and  u  a  function  of  x^ 

mudx  =  m    udx. 

Since  each  member  of  this  equation  involves  an  arbitrary 
constant,  the  equation  only  implies  that  the  two  members  have 
the  same  differential.  The  differential  of  an  integral  is  by 
definition  the  quantity  under  the  integral  sign.  Now  the 
second  member  is  the  product  of  a  constant  by  a  variable 

factor ;  hence  its  differential  \'=>md\  \udxV  that  is,  m  u dx,  which 

is  also  the  differential  of  the  first  member. 


§  T.]  ELEMENTARY   THEOREMS, 


8.  This  theorem  is  useful  not  only  in  removing  constant 
factors  from  under  the  integral  sign,  but  also  in  introducing 
such  factors  when  desired.     Thus,  given  the  integral 

recollecting  that 

d{x''-^')  -  (it  +  \)x''dx, 

we  introduce  the  constant  factor  n  +  i  under  the  integral  sign ; 
thus, 

\x''dx  —  — —  \{n  +  \)x"dx  =  — —  x"-^ '  4-  C 

9.  If  a  differential  expression  be  separated  into  parts ^  its  in- 
tegral is  the  sum  of  the  integrals  of  the  several  parts.  That  is, 
if  u^  7',  w,  '  •  '  are  functions  of  x^ 

\{ii-\-v-\-w-\-'''  ')dx  =  \ic  dx  -\-  \v  dx  +  \w dx  +  •  •  • 

For,  since  the  differential  of  a  sum  is  the  sum  of  the  differ- 
entials of  the  several  parts,  the  differential  of  the  second  mem- 
ber is  identical  with  that  of  the  first  member,  and  each  member 
involves  an  arbitrary  constant 

Thus,  for  example, 

(2  —  Vx)  dx  =   2dx  —  \xdx  =  2x  —  ^x^-\-  Cy 

the  last  term  being  integrated  by  means  of  the  formula  deduced 
in  Art.  8. 

jFundamenla/  Integrals, 

10.  The  integrals  whose  values  are  given  below  are  called 
the  fimdamental  integrals.  The  constants  of  integration  are 
generally  omitted  for  convenience. 


ELEMENTARY  METHODS  OF  INTEGRATION,   [Art.  lO. 


Formula  (a)  is  given  in  two  forms,  the  first  of  which  is  de- 
rived in  Art.  8,  while  the  second  is  simply  the  result  of  putting 
n——m.  It  is  to  be  noticed  that  this  formula  gives  an  indeter- 
minate result  when  n—  —  \\  but  in  this  case,  formula  {U)  may 
be  employed.* 

The  remaining  formulas  are  derived  directly  from  the  for- 
mulas for  differentiation;  except  that  (y'),  (^'),  (/'),  and   {in') 

X 

are  derived  from  (y),  {k),  (/),  and  {nt)  by  substituting  -  for  x. 


n-V\    ^     Jx'"  {m  —  i)  x'"-^        ^^        ^  ^ 

=  log(±^)t.-^.(^ .  .  w 

i^<    H-^^.< w 


ax 

X 


a^'dx 


cos6  de  =  sin  6*  -<-   L-     .     . 

Sin^^^rrz  _  COS  ^  .4*  .(L.     . 


(d) 


*  Applying  fonnula  {a)  to  the  definite  integral    x^dx^  we  have 

]a 

)b                     A«  +  'f          ^«  +  I 
x'^dx^^- Hf , 
a                       n+  I 

which  takes  the  form  -  when  «  —  —  i ;  but,  evaluating  in  the  usual  manner, 

= ^— ^^-  =log3-loga; 

n  +  1       J«  =  —  I  I  J«  =  —  I 

a  result  identical  with  that  obtained  by  employing  formula  (d). 

f  That  sign  is  to  be  employed  which  makes  the  logarithm  real.    See  Diff.  Calc, 
Art.  43. 


§  I.]  FUNDAMENTAL  INTEGRALS.  9 

, 

I^^H^^^^'""^--^-^ (^^ 

— ?-^-^  =    cosec  6*  cot<9^6' =  —  cosec  <9.-V:  C«^.    .     (/) 

|y(j  ^^)  =  sin-i  ;r  +  (7  =  -  cos-i  x -r  C     .    '.     .     (y) 

\    y,  2^    ji\  =  sin-^  -  +  6^=  -  cos"^-  +  C  .    .     .     (/') 
J  -/(^^  —  ;r)  a  a  ^-^  ' 

{j^^^  =  tRn-'x+ C=---cot-'x+  C,    .....     (/^) 

[-2^  =  -tan-i-+  C=  -  -cot-i-+  (:".    .    .     .     (k') 
}ar  +  ^      a  a  a  a  ^   ' 

fdx 
x^i^  -  I)  ^sec"^^  +  (7=  -  cosec-^;ir  +  C.   .     .     (/) 


lO  ELEMENTARY  METHODS  OF  INTEGRATION,       [Ex.  I. 


Examples  I. 

Find  the  values  of  the  following  integrals 
[dx 

i  [dx 

,  {   dx 

'  h  x^ 

r  dx 

/    5.        Vxdx, 
^      6.    [    (x-  ifdx, 

a 

yj   7.    I    (a  —  bxYdXy 

\      8.  J     {a  +  x^dx, 
f^'dx 

[~^dx 


2Vx. 


Vx' 


X^   +  X  —  ^. 

3 


a'^x  —  abx  4- 


3  Jo  ~  Zb' 
2  log  ^. 


log  (  -  x) 


=  log  2. 


I.] 


EXAMPLES. 


II 


./  II.  --^dXy 

J  12.  t'dXy 

-^13.  I  sin  6^9, 

•'  o 

x/  14.  COSOT^X, 

5-  J„  cos'  61' 

■  Jo    4/(«'-.v')' 


2  4/a:(d!^  +  l^jc  +  ^x"")  I  =  23!-^  •  a'\ 


f^  —  I. 


I  —  cos  c. 


sin  .r 


tan/9 


=  o. 


irr 


X~]i-^         7t 

sin-^  -  1    =  -  . 


>/     19.  If  a  body  is  projected  vertically  upward,  its  velocity  after  t  units 
of  time  is  expressed  by 

a  denoting  the  initial  velocity  ;  find  the  space  .fi  described  in  the  time 
U  and  the  greatest  height  to  which  the  body  will  rise. 


\ 


^^   =\    1)  dt  —  at,    —  -kg^r^y 


,  .a  a 

when  V  —  o  ,f=  ~~,s  =  — 

i^  2g 


Kk- 


i^ 


12  ELEMENTARY  METHODS   OF  INTEGRATION,    [Ex.  L 

N     20.  If  the  velocity  of  a  pendulum  is  expressed  by 


nt 
V  =^  a  cos  — 


the  position  corresponding  to  /  =  o  being  taken  as  origin,  find  an  ex- 
pression for  its  position  s  at  the  time  t,  and  the  extreme  positive  and 
negative  values  of  j. 


2ra    .    7tt 

s  =  — —  sm  — 
TT  2r 


s  =  ± when  f=T,  ^r,  ^t,  etc. 


I 

21.  Find  the  area  included  between  the  axis  of  x  and  a  branch  of 
the  curve 

y  —  sin  X.  2. 

[      22.  Show  that  the  area  between  the  axis  of  x,  the  parabola 

and  any  ordinate  is  two  thirds  of  the  rectangle  whose  sides  are  the 
ordinate  and  the  corresponding  abscissa. 

A      23.  Find  (a)  the  area  included  by  the  axes,  the  curve 

and  the  ordinate  corresponding  to  .^c  =  i,  and  (/3)  the  whole  area  be- 
tween the  curve  and  axes  on  the  left  of  the  axis  oiy. 

{a)  8  -  I,  (/?)  I. 
I    24.  Find  the  area  between  the  parabola  of  the  nth  degree, 

and  the  coordinates  of  the  point  (a,  a).  , 

»  +  I 


§  I.]  EXAMPLES.  13 

^25.  Show   that   the   area   between   the   axis  of  .r,  the  rectangular 

hyperbola 

xy—i, 

the  ordinate  corresponding  to  :r  ==  i,  and  any  other  ordinate  is 
equivalent  to  the  Napierian  logarithm  of  the  abscissa  of  the  latter 
ordinate. 

For  this   reason   Napierian   logarithms  are   often   called  hyperbolic 
logarithms. 

J       26.  Find  the  whole  area  between  the  axes,  the  curve 

and  the  ordinate  for  x  =.  a^m  and  n  being  positive. 

li  n>  m. 


na 


n  —  m 
if  n^m^  C50. 

/    27.  If  the  ordinate  BR  of  any  point  B  on  the  circle 

x^^f^a' 

be  produced  so  that  BR  •  RP  =  a"^,  prove  that  the  whole  area  between 
the  locus  of  R  and  its  asymptotes  is  double  the  area  of  the  circle. 

J      28.  Find  the  whole  area  between  the  axis  of  x  and  the  curve 

y  (a'  +  x')  =  a\ 

7ta\ 

29.  Find  the  area  between  the  axis  of  x  and  one  branch  of  the  com- 
panion to  the  cycloid,  the  equations  of  which  are 

xz=zaip  y  =  a  (i  —  costp). 


14  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  II. 


II. 

Direct  Integration. 

II.  In  any  one  of  the  formulas  of  Art.  lo,  we  may  of  course 
substitute  for  x  and  dx  any  function  of  x  and  its  differential. 
For  instance,  if  in  formula  (b)  we  put  x  —  am.  place  of  x^  we 
have 

J  -^~a  ^  ^^^  ^^'  ~  ^^         ^^         ^^^  ^^  ~  ^^' 

according  as  x  is  greater  or  less  than  a. 

When  a  given  integral  is  obviously  the  result  of  such  a  sub- 
stitution in  one  of  the  fundamental  integrals,  or  can  be  made 
to  take  this  form  by  the  introduction  of  a  constant  factor,  it  is 

said  to  be  directly  integrable.     Thus,     sin  7;^  ;r  ^jr  is  directly  in- 

tegrable  by  formula  {e)  ;  for,  if  in  this  formula  we  put  mx  for  B, 
we  have 

j  sin  nix  •  7ndx  =  —  cos  mx , 

hence 

sin  mx •  mdj;  =  —  —  cos  mx. 


I  sin  mx  dx  —  — 
J  ///  J 

So  also  in  ^  4/(^  4-  b^)  x  dx , 


m 


the  quantity  x  dx  becomes  the  differential  of  the  binomial 
{a  +  bx^)  when  we  introduce  the  constant  factor  2/^,  hence  this 
integral  can  be  converted  into  the  result  obtained  by  putting 


(a  +  b:^^  in  place  of  ;i:  in    j^  xdx^  which  is  a  case  of  formula  {a), 

lUS 

['^{a^b:^)xdx  =  —-\{a-V  b:^f  2bx  dx  = -~(a  +  bx'f . 


Thus 


§11.]  DIRECT  INTEGRATION,  1$ 

12.  A  simple  algebraic  or  trigonometric  transformation 
sometimes  suffices  to  render  an  expression  directly  integrable, 
or  to  separate  it  into  directly  integrable  parts.  Thus,  since 
—  sin  X  dx  is  the  differential  of  cos  x,  we  have  by  formula  ifi) 

f  ,         ( sin  X  dx  , 

tan  X  dx  =     =  —  log  cos  x . 

J  J     cos  ^ 

So  also,  by  formula  (/),  ' 

\x2.n^  ddS  ^[{s^ee-  I)  ^^=  tan  (9 -6*; 

by  (e)  and  {a), 

fsin^  Ode  =  j(i  -  cos2 6)  sin  6'^6>  =  -  cos  ^  +  j  cos^ ^ :       "'         \^ 

by  (j)  and  (a), 

"^1^(1  -;^)-^j^^  -4:^)-^(-2^^^)  =  sin-^r-  1/(1  -^. 

Rational  Fractions.  "^  \    \/7--^ 

13.  When  the  coefficient  of  dx  in  an  integral  is  a  fraction 
whose  terms  are  rational  functions  of  x^  the  integral  may  gen- 
erally be  separated  into  parts  directly  integrable.  If  the  de- 
nominator is  of  the  first  degree,  we  proceed  as  in  the  following 
example. 

Given  the  integral       y  —  ^ ^dx\ 

^  J     2;ir  +  I 

by  division, 

2x  +  1         2      4       4  2^  -f  i' 


1 6  ELEMENTARY  METHODS  OF  INTEGRATION.  [Art.  1 3c 

hence 

dx 


'^-'*id,=.'-\,d.^nd,*^^ 


H--y 


2x  +  I  2J  4J  4  J 


2;ir  +  I 


4        4        o 

When  the  denominator  is  of  higher  degree,  it  is  evident  that 
we  may,  by  division,  make  the  integration  depend  upon  that  of 
a  fraction  in  which  the  degree  of  the  numerator  is  lower  than 
that  of  the  denominator  by  at  least  a  unit.  We  shall  consider 
therefore  fractions  of  this  form  only. 


Denominators  of  the  Second  Degree. 

14.  If  the  denominator  is  of  the  second  degree,  it  will  (after 
removing  a  constant,  if  necessary)  either  be  the  square  of  an 
expression  of  the  first  degree,  or  else  such  a  square  increased 
or  diminished  by  a  constant.  As  an  example  of  the  first  case, 
let  us  take 

The  fraction  may  be  decomposed  thus : 

X  -V   \  X  —  \  -\-  2  I  2 


(x  -  if  ~    (x  -if    -  x-i^  (x-  if  ' 
hence 

[   X  +  I      J        [    dx      ^       [      dx 

=  log  {x  -  I) 
(6.  The  integral  f    .  ^  "^  ^    .  dx 


§11.]  DENOMINATORS  OF   THE   SECOND  DEGREE.  1/ 

affords  an  example  of  the  second   case,   for  the  denominator 
may  be  written  in  the  form 

x^  -V  2x  -\-  6  =  {x  -^  \f  4-  5. 

Decomposing  the  fraction  as  in  the  preceding  article, 

•^  +  3       _       X  ^\  2  ^^X-^ij    c^- 

whence  >.  ' 

[       ^  +  3       ^^  ,f(^+  ^)dx  [  dx  ^ 

The  first  of  the  integrals  in  the  second  member  is  directly 
integrable  by  formula  {b),  since  the  differential  of  the  denom- 
inator is  2  (;r  +  \)dx^  and  the  second  is  a  case  of  formula  (k'). 
Therefore  »    I*" 

(X  ■\-  X  2  X  •\-  \  ^  '^ 

-2  .    ^      .    ^^-y  =  i  log  V-^'^  +  2;ir  +  6)  +  -—  tan-^  — -—  . 
;r  +  2;f  +  6  ^      ^  \  /         y^  yg 

c 

16.  To  illustrate  the  third  case,  let  us  take 

f    2.r  +  I 


\x'-  X  -6 


dxy 


in  which  the  denominator  is  equivalent  to  (x  —  yf  —  6^,  and 
can  therefore  be  resolved  into  real  factors  of  the  first  degree. 
We  can  then  decompose  the  fraction  into  fractions  having  these 
factors  for  denominators.  Thus,  in  the  present  example,  as- 
sume 

2;ir  +  I  A  B 


x^  —  X  —6      X—  ^      X  +  2 


(0 


in  which  A  and  B  are  numerical  quantities  to  he  determined* 
Multiplying  by  {x  —  3)  (x  ~\-  2), 

2X+  I  =A{x  -{-  2)  -{-  B{x-^. (2) 


1 8  ELEMENTARY  METHODS   OF  INTEGRATION.  [Art.  1 6. 

Since  equation  (2)  is  an  algebraic  identity,  we  may  in  it  assign 
any  value  we  choose  to  x.     Putting  ;tr  =  3,  we  find 


7  =  5^, 

whence 

A=h 

putting  X  — 

-2, 

' 

-  3  =  -  5^, 

whence 

B  =  \.' 

Substituting 

'  these  values  in 

(I). 

2X  +   I 

5(^-3)^ 

5(^ 

3          ■      .1 

;,^_^_6- 

•  +  2)'        / 

whence 

f      2X  ^-    \ 


^dx  =  I  log  (x  -  3)  +  I  log  {x  +  2). 

17.  If  the  denominator,  in  a  case  of  the  kind  last  considered, 
is  denoted  by  (x  —  a)  (x  —  d),  a  and  b  are  evidently  the  roots  of 
the  equation  formed  by  putting  this  denominator  equal  to  zero. 
The  cases  considered  in  Art.  14  and  Art.  15  are  respectively 
those  in  which  the  roots  of  this  equation  are  equal,  and  those 
in  which  the  roots  are  imaginary.  When  the  roots  are  real  and 
unequal,  if  the  numerator  does  not  contain  x^  the  integral  can 
be  reduced  to  the  form 

f  dx 

]{x-d){x-by 

and  by  the  method  given  in  the  preceding  article  we  find 

f  dx  T 


\x  —  a)  {x  —  b) 


log  {x  -  a)  -  log  {x  -  b)\ 


'<-^y ^^y 


*  The  formulas  of  this  series  are  collected  together  at  the  end  of  Chapter  II., 
for  convenience  of  reference.     See  Art.  loi. 


§  II.]  DENOMIN-A  TORS  OF   THE   SECOND  -DEGREE.  IQ 

in  which,  when  x  <  a,  log  {a  —  x)  should  be  written  in  place  of 
log  {x  —  a).     [See  note  on  formula  (b),  Art.  lo.] 
If  ^  =  —  «,  this  formula  becomes 

f_^  =  ±iog^:::f ^a') 

}x^  —  a^      2a     ^  X  -{-  a  ^     ^ 

Integrals  of  the  special  forms  given  in  (A)  and  (A')  may  be 
evaluated  by  the  direct  application  of  these  formulas.  Thus, 
given  the  integral 

f  ^^ 

}2x^  -{-  ^x  —  2' 

if  we  place  the  denominator  equal  to  zero,  we  have  the  roots 
a  =  ^,  d  =  —  2;  whence  by  formula  (A), 


dx 


2^  +  3.r  —  2 


dx  _  I       I    -       X  —  \  ^ 

(.tr—  I)  (JT2)  ~  2  ■  2i    °^  X  ^  2  ' 


or,  since  log  {2x  —  i)  differs  from  log  {x  —  l)  only  by  a  con- 
stant, we  may  write 

f  dx  I  -       2jr  —  I 

log 


2x^  -{-  ^x  —  2       5      ^    X  +  2 


Denominators  of  Higher  Degree. 

18.  When  the  denominator  is  of  a  degree  higher  than  the 
second,  we  may  in  like  manner  suppose  it  resolved  into  factors 
corresponding  to  the  roots  of  the  equation  formed  by  placing  it 
equal  to  zero.  The  fraction  (of  which  we  suppose  the  numerator 
to  be  lower  in  degree  than  the  denominator)  may  now  be  decom- 
posed into  partial  fractions.  If  the  roots  are  all  real  and  un- 
equal, we  assume  these  partial  fractions  as  in  Art.  16 ;  there 
being  one  assumed  fraction  for  each  factor. 

li,  however,  a  pair  of  imaginary  roots  occurs,  the  factor  cor- 


20  ELEMENTARY  METHODS  OF  INTEGRATION.  [Art.  1 8. 

responding  to  the  pair  is  of  the  form  {x  —  of  -\-  ^^,  and  the 
partial  fraction  must  be  assumed  in  the  form 


Ax  ■\-  B 

(x  -  of  +  fi 


:2' 


for  we  are  only  entitled  to  assume  that  the  numerator  of  each 
partial  fraction  is  lower  in  degree  than  its  denominator  (other- 
wise the  given  fraction  which  is  the  sum  of  the  partial  fractions 
would  not  have  this  property). 


19.  For  example,  given 


^  dx. 


][:x^  ^  \){x  -  i) 
Assume 


jr  +  3  Ax  +  B         C  ,  . 


{:>^  •¥  \)(x  —\)        x^  +  I         X  —  1 
whence 

X  +  3  =  (x  -  i){Ax  +  B)  +  {x^  +  i)  a 
Putting  X  =  ly 

4  =  2Cy  whence       C=2; 
putting  X  =  Of 

5  =  —  B  -h  Cy     whence       B  =  —  i. 

To  determine  A,  any  convenient  third  value  may  be  given 
to  X ;  for  example,  if  we  put  x  =  —  i,  we  have 

2  =  -2{-A  +  B)  -h  2C  .-.  A=^2, 

Substituting  in  (i), 

;r+3  _       2  2;ir+l 

{x^  -\-  i){x—  i)  ~  ^^^  ~  ;r^  +  I  ' 


§11].  DENOMINATORS  OF  HIGHER  DEGREE.  21 

therefore 

J;tr  +  3  J    _     [   dx  {2xdx        {    dx 

=  2  log  {x  —  i)  —  log  (ji:^  +  I)  —  tan"  ^ X. 

20.  If  the  denominator  admits  of  factors  which  are  func- 
tions of  jc^y  and  the  numerator  is  also  a  function  of  or^,  we  may 
with  advantage  first  decompose  into  fractions  having  these 
factors  for  denominators.     Thus,  given 

f  x^dx 
]x*-a*' 

Putting  J/  for  x^  in  the  fraction,  we  first  find 


hence 


^      _         I  I 


f  x^dx   _  I  (    dx  C    dx 


therefore  [see  equation  {A'),  Art.  17], 

x^dx    _ 

4a 


fx^dx          I    ,       X  —  a        I  .  ;r 

-T 1  =  —  log  — — -  +  —  tan-  ^~ , 
XT  —  or     Aa         X  -{•  a      2a  a 


This  method  may  sometimes  be  employed  when  the  nume- 
rator is  not  a  function  of  x^ ;  thus,  since 


xf'-a'-  2a\x^  -  a")      20" {x"  +  a")' 
we  have 


x^-a'      20"  {x'  -  a")      2a^  {x^  +  a^) ' 


22  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  20. 


hence 


X  dx  I    ,       x^  —  c^ 

log 


]x!'-(^     4^   ^;r^  +  ^2• 
21.  The  fraction  corresponding  to  a  pair  of  equal  roots,  that 
is,  to  a  factor  in  the  denominator  of  the  form  {x,^  dfy  is  (see 
Art.  14)  equivalent  to  a  pair  of  fractions  of  the  form 


A  B 

+ 


X  —  a      (x  —  a) 


■2 ' 


we  may,  therefore,  at  once  assume  the  partial  fractions  in  this 
form.  We  proceed  in  like  manner  when  a  higher  power  of  a 
linear  factor  occurs.     For  example,  given 


we  assume 


X  +  2  A  B        ^      C  D 

+  7 ^,  + r  + 


(x  —  if{x  +  1)       {x  —   if   '    (x  —  if   '  X  —  I    '  X  +  I 

whence 

x-h2=lA-{-B{x-  i)  +  C{x-if]{x+  i)^D{x-if.    .  (i) 

Putting  ;r  =  I,  we  have 

3  =  2A         .-.        A=i. 

The  values  of  B  and  C  may  be  determined  as  follows :  if  we 
substitute  the  value  just  determined  for  Ay  equation  (i),  is 
identically  satisfied  by  x  =  i,  hence  it  may  be  divided  by  jr  —  i. 
We  thus  obtain 

^^=[B  +  C{x-i)-](x+  i)+D{x-if  .     .     (2) 


§  IL]  MULTIPLE  ROOTS.  23 

in  which  we  may  again  put  x  =  \,  whence  B  —  —  \.     In  like 
manner  from  (2),  we  obtain 

l^C{x-^  \)^D(x-\), 

from  which  C  —^^  and  Z>  =  —  J.     Therefore 

r        x^2  J   _?>[    d,x  \  [    dx  \[  dx        \  {    dx 

J(;ir-lf(;r+i)  2j(;r-i)«~4J(;r- i)2'^8j;^^~8  J^TTl 

^  I  \  ,      X-  \ 


X 


A(x-\f      \(x  -  i)    '  8     ^;ir+  I 

22.  In  this  example,  after  obtaining  the  values  of  A  and  D 
from  equation  (i)  by  putting  ;tr  =  i,  and  x  =  —  \^  two  equations 
from  which  B  and  C  might  be  obtained  by  elimination  could 
have  been  derived  by  giving  to  x  any  two  other  values.  Con- 
venient equations  for  determining  B  and  C  may  also  be  obtained 
by  putting  ;ir  =  i  in  two  equations  successively  derived  by 
differentiation  from  the  identical  equation  (i).  In  the  first  dif- 
ferentiation we  may  reject  all  terms  containing  (x  —  if  \  since 
these  terms,  and  also  those  derived  from  them  by  the  second 
differentiation,  will  vanish  when  x  =  \,  Thus,  from  equation 
(i),  Art.  21,  we  obtain 

\  —  A  ^  2Bx  +  2C  (x^  —i)  +  terms  containing  {x  —  if. 

Putting  X  =  I,  and  ^  =  | ,  we  have  B  =  —  I.     Differentiating 
again  and  substituting  the  value  of  B, 

o  =  —  ^  -\-  4Cx  +  terms  containing  {x  —  i), 
and,  putting  x  =  i  m  this  last  equation,  C  =  \  . 

23.  When  the  method  of  differentiation  is  applied  to  a  case 


24  ELEMENTARY  METHODS  OF  INTEGRATION,     [Art.  23. 

in  which  more  than  one  multiple  root  occurs,  it  is  best  to  pro- 
ceed with  each  root  separately.     Thus  given, 


f  -y  +  I  . 

](x-  \f(x\2f' 


(x  —  \f  (x -^  2f      (x  —  if       x—i        {x  +  2f      X+2 
whence 

x+i={A+B{x-i)-]{x  +  2f-^[C+D{x  +  2)]{x-if  .   .    (I) 
Putting  ;r  =  I,  and  ;r  =  —  2,  we  derive 

A  =  '-^  C=-'-. 

9  9 

Differentiating  (i),  we  have 

I  =  2A  (x  +  2)  +  B  {x  +  2y  -^  terms  containing  {x  —  i), 

2                                I 
whence,  putting  x  =  ly  and  A  ==  - ,  we  have  B  = . 

Again,  differentiating  (i),  we  have 

I  =  2C  {x  —  i)  -i-  D  (x  —  if  +  terms  containing  (x  +  2), 
whence,  putting  x  =  —  2y  and  C  = ,  we  have  D  =  — . 

Therefore 

f X  -\-  I  _  _         2  I  J_,      X  +  2 

]{x  -  if  {x  +  2f^  ~       g{x-  l)"^  9  (;ir  +  2)  "^  2y^^x-  I  * 

24.  Instead  of  assuming  the  partial  fractions  with  undeter- 


§11.]  RATIONAL  FRACTIONS.  25 

mined  numerators,  it  is  sometimes  possible  to  proceed  more 
expeditiously  as  in  the  following  examples : 
Given 


U  (I  +  ^) 


dx\ 


putting  the  numerator  in  the  form  i  +  .r^  —  ;i:^,  we  have 


'dx_ 
,3 


x{i  +  x^) 
Treating  the  last  integral  in  like  manner, 


X  dx 
1? 


=  -^-log^  +  Jlog(i+^)=-^+log-ili^. 
Again,  given 

putting  the  numerator  in  the  form  (i  +  xf  —  2x  —  x^,  we  have 

f         I  ,    _  [dx      r    2  +  X      , 

J;^(i  ^xj'^'^-]l?-]x{Y  +xf'^'' 

_  [dx  r       dx  f      dx 

~J:?  "  ^]x{i  +  xf^iii  +xY' 

Hence  by  equation  {A),  Art.  17, 
dx 


C         dx  I  , 


+  X         I   +  X 


X 


26  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  II. 

y, 

Examples  II.  <=i^^' 

/r.f^,  -log(«-.). 


J 


fdx 
{a-xf' 


7- 


d  -  (a'  -  x')^^ 


3     ^a"--^" 


(a  —  xf^  a  —  X 

[      XdX  I     1  /     2      ,  2\ 

/      f     jv'  dx  .  ,  I , 

8.      («  +  wj;)"  dx^ 

9-  Jsin"2^'       '^^^ 

V     lo.    \  co^^  X  ?,m.x dx,     j'^^.w.,  ^j.v^'  ' 
1  f  cos  0  dB     , 

>i      12.      sec' 3  A- tan  3^  <3^r,         •^  -  (^mTW^ 


(«^  +  3^y 

24 

(^ 

4-;«a)'-«'' 

3»« 

cot  2;«' 

2 

I  —  COS*  X 

I 

2 

cosec'' 

0. 

sec' 

3-^- 

I 

9 

§11.]  EXAMPLES,  27 

/      13.  fd!'«-^^,         (Km^  ^T     • 

"^  J  ^  m\oga 

>1  14.  I  (f-^  -  \fdx,    -  -JfS-^  —  |f=^  +  3£^  -  ^. 

/  f/      ,        •  8   \9   •  ^         .  ^  (i  +  3 sin' J?)' 

V   15.      (i  +  3  sm  :r)  ^\XiXQ.O'i>xax,      \.   -^-iL  )  ^ ^. 


J  o  '^yiax  —  x^)  Jo 

v|    17.  Kos^'o^e, -^^«('---^^'^      ^^  -. 

^    18.    I  sec* Q ^9,    7'^^^^^^  A^' ^  tan0+-tan'O. 

19.     tan'^y^,  —  tan"^  +  logcos^. 

IT  n 

V   20.        sec*  X  tan  ^  ^jc,  kJU,--*.*-^                            -  sec*  ^      =  — . 

Jo  .                                            4            Jo       4 

.    [4/^  ~  ^ dx,  ezsin"^-  +   4/(^' —  ^'). 

jy  a  +  X     '  a            ^ 

J  Z  2 
4 

|/ ^jf,  i^{2ax  —  x"*)  -\-  a  vers "  ^  - . 

•          f  .    /             N  ^  cos(«—  29) 

24,        sin  {OL  —  20)  ^0,        ^.SU^^e^y-  • 


21. 


28  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  II. 


25 


f    CO! 


cos  X  dx 


V 


''■  l: 


^  sin  ^  * 
dx 


tan.r  * 


■7t>«^ 


—  -7  log  (/2  —  ^  sin  J^). 


i  log  2. 


^'^'   J  „  tan  .r ' 


i  log  2. 


r- 


log  (-  log.T)j'  =  -log 

4 


tan-  ^f^. 


^  30.  j^.  r-^' 

^33.}^,     .^^i^V 


^34.  j;:,,,^,?^^,  r%^^  ^ 


v/35.  fprj^q- 


V3 


tan" 


3 

tan  -  'x\ 

I 
2 

sm    ^^. 

I 

4^3 

sin" 

.1  -^4/3 
i/5    * 

I 
4/10 

r 

tan" 

1  ^Vs 
4/2   • 

4  ' 

2.Jk:  + 

-11' 

TT 

zVi 


" .  w.v,^y^ 


^11.] 


EXAMPLES. 


dx 


v(s  -4^-  ^y     I  ^^y^ 


^ 


/    3.r 


V"^  29 

cos"^  f . 


'V^v. 


/    38.  P'-J^yr,    J^^ 

J  o  H  ~~"  AT 

V     39.  ]      .,^  +  I       ^> 


fJi:'  +^  4-  I   ^        I 
4^-   J^^  _^  4-  i^'>    -*■  +  ^^g  (-^^    -  ^'  +  i)  + 


VC^''  —  a'')—  a  sec  ~  *  - . 


a'  (log  2  -  t). 


4^  —  J  log  (a:'  +  i)  —  tan  ■  ^ ^'. 


2  ,  2^1'  —   I 

tan"' 


Vs 


Vs 


41. 


a:  +^log — — 

4     °  Jt'  +  2 


i°g(r^-^--"- 


/43.  Ji^^^^.  ^bXi^X'         --  -^  +  ^  log  (^^  +  3). 
^    45-  J^.,.^.,.^^'^-^,       ■         ■  ^  +  Iog 


X  —  I 


46 


r 

J  o 


dx 


t,^..JrXC^ 


X  —  2ax  cos  a  -\r  a 


a  sm 


,  .V  —  <3!  COS  or\  ^        7C  —  a 

—  tan  '  ^ -. = -. 

OL  ^  Sin  ^  2a  sm  a 

— lo 


30  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  11. 


t'   47. 
48. 
49. 
\/      50. 


1.^ 


dx 


2ax  sec  «  +  « 


y 


51- 


52. 


53. 


\y  54. 

/ 

V    55- 

^J 


56. 


/ 


M    57- 


58. 


f  dx 

J  2.v'  —4^—7' 

[•  x"  dx 
\i-x'" 

J  x"  —  jt""  —  2J«;      ' 

J  (^  +  2)  (^  +  3)"' 

x^^  x"  -^  x^  i' 
J  ^^  +  ^«  _  2  » 

f  ^'  —  jr  +  2    , 
J  :<:*  -  5^'  +  4 

r  ^^ 

]x'  -  x^  -  X  ^  y' 


X  —  a  sec  «:  —  ^  tan  or 


2a  tan  a      °  :f  —  «  sec  a  -\-  a  tan  a; 


log 


^2     ,         2^  —   2   —   3  4/2 

12        ^  2Jt:  —  2  +  3  4/2' 


I ,     I  +  ^ 


2  log 


:r  +  2      ^  +  3 


-[_tan    ^^  +  log     ^^^     J. 


7  log — ■ —  +-^— tan 
6     ^ x-^-  \         3 


1/2  • 


log 


2,       ^+11-       X—  2 

-log ; +  -log 

3       ^X-V2         3       ^X-\ 


I  .        Jg  +   I I 

4     °^  —  I        2{x  —  l) 


2 


(X^  +   l)^ 


I  ,          (^  +   l)'       ,        I      ^       _i2:t:  —   I 
-log-^^ '- — H ;— tan    ' — — 

6     ^^  —  ^  +  I         4/3  4/3 


1(^^#FT7)'    7iog(— i)-^iog(^'  +  x)-^). 


§  II.]  EXAMPLES. 


"^■^ 


log  «  —  -  log  (i  +  a;) log  (i  +  a;') tan  - '  x. 


6'-  jJ^+^/_6:,'^^'  ilogx  +  ilog(^-2)+ilog(*  +  3). 
/     ,       f         x^dx  \  ^      X  —  2         \/  X  X 

^       62.      \- T, ,  -log ; +     "^  ^  —-- 

}x'  —  X'  —  12'  J       ^  X  -h  2 


tan- 


7     ^  X  -h  2  7  V3' 


J 


f      xVjc  I         ^  —  I                 ^ 

^-  ](x'-iY'  4  ^^^r+7"  2(x'^- 1)* 

i    '64.    f^4^^.,  Xtan-^^--i-log^. 

^        (         xdx  \  .        X^  —  2 

(/                   ["                      ^^  7t 

V       J^x'Ca"  +  xy  '                    ■                  41  ■ 

\l           [        dx  n     \^                                                 .11 


32  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  11. 

/  „   f ^^ \^v. 

/   ,,     f ^ ?A-- 

*     "•  ]x'{a^bx'y 

J    74.  Find  the  whole  area  enclosed  by  both  loops  of  the  curve 

yj     75.  Find  the  area  enclosed  between  the  asymptote  corresponding 
to  X  =  a,  and  the  curve 


lr»rr  _ 

X 

1 

I 

log- 

i+X    ' 

I  +x' 

i'°s« 

x' 
+  bx" 

I          , 

^  ,    d? 

+  dx' 

2ax^ 

^^"'"^ 

x'       ' 

J 

\ 


2       2        •  2      2  2       2 

x"  y"  +  ax   —  ay. 


76.  Find  the  whole  area  enclosed  by  the  curve 

a'f  =  x'  {a-  -  x'). 

77.  Find  the  area  enclosed  by  the  catenary 


the  axes  and  any  ordinate. 


£^    -H  £ 


']■ 


^[•-•-•} 


78.  Find  the  whole  area  between  the  witch 

.ly  =  4a''  {2a  —  x) 

and  its  asymptote.     See  Ex.  23. 


4;ra^ 


§  III.]  TRIGONOMETRIC  INTEGRALS,  33 


III. 

Trigonometric  Integrals.  ~  - 

25.    The    transformation,     tan'^^  =  sec^  ^  —  I,    suffices    to 
separate  all  integrals  of  the  form 

Itan^ede, (I) 

in  which  n  is  an  integer,  into  directly  integrable  parts.     Thus, 
for  example, 

ftan«  ede  =  [tanS  6  (sec^  6  -  i)  dd 

_  tan^  6       i 
~~4  J 


^^"'^       't^n^Bde, 


Transforming  the  last  integral  in  like  manner,  we  have 

r.     'i  n  7n      tan*  0      tan^  6      r         .   ,  „ 
tan^<9^6>=: +   tan  OdO; 

hence  (see  Art.  12) 

L     nn^n      tan*<9      tan2(9      ,  . 

tan^  Odd  = log  cos  6. 

When  the  value  of  n  in  (i)  is  even,  the  value  of  the  final  inte- 
gral will  be  6,     When  n  is  negative,  the  integral  takes  the  form 

[cot"^^^', 
which  may  be  treated  in  a  similar  manner. 


34  ELEMENTARY  METHODS  OF  INTEGRATION.     [Art.  26. 

26.  Integrals  of  the  form 

[sec«(9^(9 (2) 

are  readily  evaluated  when  n  is  an  even  number,  thus 
[s^eedB  =  [(tan^  +  i)2  sec'ede 

=  [tan*  ^  sec2  6  dS  +  2  [tan^  ^  st^ddd  +  [sec^  ^  ^^ 

tan»  6*      2  tan^  (9 

=  —r—  + ;; +  tan  6. 

5  3 

If  ft  in  expression  (2)  is  odd,  the  method  to  be  explained  in 
Section  VI  is  required. 

Integrals  of  the  form     cosec*^6d6  are  treated  in  like  manner. 

Cases  in  which  sin'^  Q  cos''  Q  dd  is  directly  integrable, 

27.  If  n  is  2.  positive  odd  number,  an  integral  of  the  form 

I  sin'^  d  cos«  Q  dB (3) 

is  directly  integrable  in  terms  of  sin  B,     Thus, 

[sin^  B  cos^  BdB=  [sin^  ^  (i  -  sin*  6')2cos  Bdd 

_  sin^  B      2  sin^  B      sin^  B 
~"T~  5       "^      7     ' 

This  method  is  evidently  applicable  even  when  m  is  frac- 
tional or  negative.     Thus,  putting  ;y  for  sin  B, 


§  III.]  TRIGONOMETRIC  INTEGRALS.  35 


^m«-f^-\>-*"-V'r. 


■cos'^  j^_  f(i  -J)dy 

y\ 

hence 

•cos^  e  ^^  i2i  23  +  sin^  e 


f  COS^  ^  1         2     i  2 

Jsint6l  -^  r  'h 


3-"  3        i/(sin^)- 

When  m  in  expression  (3)  is  a  positive  odd  number,  the  in- 
tegral is  evaluated  in  a  similar  manner. 

28,  An  integral  of  the  form  (3)  is  also  directly  integrable 
when  m  +  n  /j  an  even  negative  integer^  in  other  words,  when  it 
can  be  written  in  the  form 


J  cos'«+^^  d      J 


in  which  q  is  positive. 
For  example, 

dd 


fad  f 

^-r^ r^  =  (tan  ^-^  sec*  6  dd 
sma  ^  COS8  ^       J  ^  ^ 

=  f(tan  (9)-t  (tan^  ^  +  i)  sec^ddd ; 


hence 


Jsma 


=  -tant^ 


^cos^^      3  tan«^ 


It  may  be  more  convenient  to  express  the  integral  in  terms 
of  cot  6  and  cosec  6,  thus 

i^^i^  =  jcot*  d  (cot2  ^  +  I)  cosec'ede 

cot"^  e      cot«  (9 


36  ELEMENTARY  METHODS  OF  INTEGRATION.     [Art.  28. 

Integrals  of  the  forms  treated  in  Art.  25  and  Art.  26  are  in- 
cluded in  the  general  form  (3),  Art.  27.  Except  in  the  cases 
already  considered,  and  in  the  special  cases  given  below,  the 
method  of  reduction  given  in  Section  VI  is  required  in  the 
evaluation  of  integrals  of  this  form. 


The  Integrals  sln^  e  dd,  and  cos^  d  dd. 

29.  These  integrals  are  readily  evaluated  by  means  of  the 
transformations 

sin^  6  =  ^{i  —  cos  2d),       and       cos^  ^^  =  i(^  +  cos  26), 
Thus 

[  sm^Odd  =  ^{dd-'\  [cos  2ddd  =  ^d  -ism  28, 

or,  since  sin  26  =  2  sin  6  cos  ^, 

[sin2  dd6  =  i(d  -  sin  8  cos  d) {B) 

In  like  manner 

\cos^  d do  =  ^{6 +  sm  6 cos  6) {C) 

Since  sin^  6  +  cos^  6  =  i,  the  sum  of  these  integrals  is  \d6\  ac- 
cordingly we  find  the  sum  of  their  values  to  be  6. 

In  the  applications  of  the  Integral  Calculus,  these  integrals 
frequently  occur  with  the  limits  o  and  ^n  ;  from  {B)  and  {C) 
we  derive 


IT  TI 

f'sin2^^^=|'cos2^^^  =  i;r. 


§111.]  TRTGO.VOME  TRIG  IN TEGRA LS.  37 


The  Integrals  \^^^^,  j^^,  and  J^. 
30.  We  have 

f     de        [speeds    ,    ,    .  ,^, 

^— 5 2)  =   —^ — z~  =  log  tan  6.     ,     .     ,    (D) 

J  sin  6  cos  6     J    tan  ^  ^  ^    ^ 

Again,  using  the  transformation, 

sin  ^  =  2  sin  ^6  cos  J^, 
we  have 

Jsin6/~Jsini^cosi^~J     tan  J^     ' 
hence 

|^,=  logta„ift (E) 

This  integral  may  also  be  evaluated  thus, 
de        [  smddd      f  sin6>^^ 


f  d6    _[•  sm  ddd       [2 
Jsin^~J    sin^^    '"U 


cos^  6  * 


Since  sin  Odd  =  —  d{cos  6),  the  value  of  the  last  integral  is,  by 
formula  {A'),  Art.  17, 

I ,      I  —  cos  ^     ,        /I  —  cos  B 


I ,      I  —  cos  c/     I        /I  — 


cos^* 


and,  multiplying  both  terms  of  the  fraction  by  I  —  cos  6,  we 
have 

{  dS        ,       I  —  cos  6  ,  r^,. 


38  ELEMENTARY  METHODS  OF  INTEGRATION,    [Art.  3 1. 

31.  Since  cos  6  —  sin  {^n  +  ^),  we  derive  from  formula  (E), 

J  cos<9      Jsm(i;r  +  <9j         ^         L4      2J  ^    ^ 

By  employing  a  process  similar  to  that  used  in  deriving  for- 
mula {E),  we  have  also 

[    dS         .      I  +  sin  <9 


Miscellaneous   Trigonometric  Integrals, 

32.  A  trigonometric  integral  may  sometimes  be  reduced, 
by  means  of  the  formulas  for  trigonometric  transformation,  to 
one  of  the  forms  integrated  in  the  preceding  articles.  For 
example,  let  us  take  the  integral 

de 


f ^ 

J  ^  sin  ^  + 


<^  cos  d* 


Putting  a^kQosa,  b  —  k  'saw  a,    ,     .     .     .     (i) 

we  have 

[•  dS i_  f       dB 

J  <a:  sin  ^  +  ^  cos  ^  ~  ^  J  sin  (l9  +  ol) 

Hence  by  formula  (E) 

J^sin^+  ^cos6>      k    ^         2^    ^    " 
or,  since  equations  (i)  give 

;^=.  V(^2  +  ^),  tan«  =  -, 

f J^ ,  =  -,    '     ,,log  tan  i  fe  +  tan-  ■:*1 . 

J  a  sin  6*  +  i^  cos  6*      */(a^  ^  I?)    ^         2L  «J 


§  III.]    MISCELLANEOUS   TRIGONOMETRIC  INTEGRALS.  39 

33.  The  expression  sin  ntd  sin  nQ  dd  may  be  integrated  by- 
means  of  the  formula    * 

cos  {m  —  n)  6  —  cos  {m  +  n)  6  —  2  sm  md  sin  nd  ; 
whence 

\smmdsmnddd  =  — ) { } f- .     .    (i) 

J  2{m  —  n)  2  {in  -{-  n)  ^  ' 

In  like  manner,  from 

cos  (m  —  n)  6  -{■  cos  {m  +  n)  6  =  2  cos  mO  cos  nO, 

we  derive 

f  X.  /.  //,      sin  (m  —  71)  6      sin  (m  -V  n)d  ,  . 

\cosmecosn6dd  = \ ^+ ) (—.    .    (2) 

J  2{m  —  11)  2  {m  +  n) 

When  m  =  fty  the  first  term  of  the  second  member  of  each 
of  these  equations  takes  an  indeterminate  form.  Evaluating 
this  term,  we  have 

sm^nddd  = , (3) 

J  f      2    /I  -7/1      ^      sin  2nO  f  n 

and  \q.o%^ nd dS  =  -  ■{ (4) 

J  24/2 

Using  the  limits  o  and  n  we  have,  from  (i)  and  (2),  zdlan.  m 
and  n  are  unequal  integers^ 

sin  w^sin;^^^^  =      cos  mS  cos  716  dd  =  O',  ..    .    (5) 

Jo  Jo 

but,  when  in  and  n  are  equal  integers,  we  have  from:  (3;)  and  (4) 
[ sm^ nddd  =  [cos'' ndde  =- .    .     ..    .    ,.    (6) 

Jo  Jo"  2 

34.  To  integrate  4/(1  4-  cos  6)  dB;  we  use  the  formula 

2  cos^  ^6  =  L  -h  cos.  6, 


40  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  34. 

whence  i^(i  +  cos  ^)  =  ±  V2cosi^, 

in  which  the  positive  sign  is  to  be  taken,  provided  the  value  of 
B  is  between  o  and  n.     Supposing  this  to  be  the  case,  we  have 

'   [  V  (I  +  cos  (9)  dQ  =  ^/2  [cos  \ede 

=  24/2  sin  \^. 

For  example,  we  have  the  definite  integral 
It 

[^  4/(1  +  cos  d)dQ  —  24/2  sin- =  2. 
Jo  4 

Intezration  of 7 -r.* 

^  -^    a  +  0  cos  t/ 

35.  By  means  of  the  formulas 

I  =cos2{,^  +  sinH^        and        cos  ^  =  cosH^  -  sin^^^, 

we  have 

f       dd         _  f de 

Jrt  +  ^  cos  ^  ~  J  (^  +  b)  cos2  4/9  +  {a  -  b)  sin^  ^8' 

Multiplying  numerator  and  denominator  by  sec^^^,  this  be* 

comes 

f  sec^idde 

]a-h  b  +  {a-b)t3in'ie' 

and,  putting  for  abbreviation 

tan  ^6  =  y, 

we  have,  since  |  sec^  1-6  d6  =  dy, 

[_      de_^     ^  2  [  dy 

J^  +  1^  cos  6*  ]a  ■\-  b  ^{a  -  b)f' 


§111.]    MISCELLANEOUS    TRIGONOMETRIC  INTEGRALS.  4I 

The  form  of  this  integral  depends  upon  the  relative  values 
of  a  and  b.  Assuming  a  to  be  positive,  if  b^  which  may  be 
either  positive  or  negative,  is  numerically  less  than  a^  we  may 
put 

a  —  0 

The  integral  may  then  be  written  in  the  form 

2      f      dy 
a-bjc"  +  f' 

the  value  of  which  is,  by  formula  {k')y 

c{a  —  b)  c 

Hence,  substituting  their  values  for  y  and  c,  we  have,  in  this 
case, 

f— 4^=-7-^-^tan-4V'^t^^4^]-    .   (Q 
]a-^  b  cos  Q      ^(a^  —  1^)  \J  a -V  b        ^  J 

If,  on  the  other  hand,  b  is  numerically  greater  than  a,  this 
expression  ior  the  integral  involves  imaginary  quantities ;  but 
putting 

b  ^  a  _   « 

the  integral  becomes 

dy 


_  f     dy 


,  b 
the  value  of  which  is,  by  formula  .(^'),  Art.  17, 

c{b  —  d)     ^c-y 


) 


42  ELEMENTARY  METHODS  OF  INTEGRATION.     [Art.  35. 

Therefore,  in  this  case, 
f        de         _  I  sf(b^d)-^W{b-d)  tan  \Q 

36.  U  e  <  I,  formula  (G)  of  the  preceding  article  gives 

f 23= -7^^— ^tan-^r^/^^tanl /9    .    .    (i) 

J I  +  ^  cos  <9      V  {i  —  e^)  [_y  1  +  e       ^    J         ^  ^ 

Putting 

|/^^-tani^=tani0, (2) 

and  noticing  that  #  =  o  when  ^  =  o,  we  may  write 

Now,  if  in  equation  (i)  we  put  ^  for  6  and  change  the  sign  of 
Cf  we  obtain 

f  —J^  =  —A^  tan-  [^/'L±^  tan  i  ^1 ; 
J^  I  —  ^  cos  ^       V{i  -  e^)  [Jy    I  —  e  ^  ^J' 

hence,  by  equation  (2), 

f           ^^          -           ^                      ■  /,^ 

J^i-^cos^~  i/(i-^^) ^^^ 

Equations  (3)  and  (4)  are  equivalent  to 

de         ^        d^ 
i+^cos^       |/(i-^)' ^5^ 

,  d^  de  .^. 

and  T  =  -77 2^ , (o) 


§  III.]  .  TRIGONOMETRIC  INTEGRALS.  43 

the  product  of  which  gives 

(i  +  ^  cos  ^)  (i  —  ^  cos  ^)  =  I  —  ^  .     .     .     .     (7) 
By  means  of  these  relations  any  expression  of  the  form 
f  dS 

J(i  +  e  zo^ey 

where  ;2  is  a  positive  integer,  may  be  reduced  to  an  integrable 
form.     For 

f  dS f        de I  . 

J(i  +  ^  cos^)'^  ~  Ji  +  ^  cos6>  (i  +  ^  cos(?)«-^  * 

hence,  by  equations  (5)  and  (7), 

dd 

e  cos 


Jji  +  ^cos^)«-(i-^)«-*J,^^ 


By  expanding  (\  —  e  cos  ^)'*~%  the  last  expression  is  reduced 
to  a  series  of  integrals  involving  powers  of  cos  ^ ;  these  may 
be  evaluated  by  the  methods  given  in  this  section  and  Section 
VI,  and  the  results  expressed  in  terms  of  d  by  means  of  equa- 
tion (2)  or  of  equation  (7). 

Examples  III. 

.         ,  tan^  mx      tan  mx 

tan  mx  ax,  — 1-  x. 

^m  m 


tan'  xdxj  1^  ~  i  log  2. 

y^.    fsec*  (6  +  a)  de,  *^'^"  "^  "^  +  tan  (6  +  a). 


44  ELEMENTARY  METHODS  OF  INTEGRATION.   [Ex.  III. 

/    r-  ^ 

Jo 

vs.     sin' 6  cos' 0  ^9, 
"/  6.      |/(sin  e)  cos'  9  ^0, 

tr 

\/ 7.    I    COS*  0  sin'  0  ^0, 

,  /       f  sin'  0  dTo  5.  i 

/       f ^ 


2.3.  4.3:         ,      2       .    il  ^ 

-  sin^0  —  -  sin2  B  -\ sm  ^  0. 

^  7  II 


2 
35 


f         dB 

9.     ^^ ^— ,        Multiply  by  sin'  0  +  cos'  0.        tan  0  —  cot  0. 

J  sin  0  cos  0 


/ 


(sm  X    , 
T-  aXy 
cos  X 


10.     — r-^/jtr.  See  Art.  2Z. 


tan*  jr 


^'•IS^T^'  i(tan'e-cot'e)  +  alogtane. 

1     I..  (i^(!'^,  Itante. 

'  J  C0S2  0 

J  f  sin'jc  dx 

^^'  J    cos''^    ' 

J  f  sin'jv  //jc 

'4-  J-WIT' 


5  cos'  ^      3  cos'  X 


tan'  a:      tan'  x 


5  3 


j       15.    |sin'0cos'0A  3V  [29 -sin  20  cos  20]. 


§  III.] 

7  ^ 

V  i6.         'siri  mx  dx, 


EXAMPLES. 


45 


n 
2fn 


v/  '^  J 


sm  e  m 

COS0       ' 


/    ,8.    pS^t^, 

Ji:     sine    ' 

3 

•I 


Sin  0  +  cos  9 
+  cos  X  ' 


log  tan    — I —     —  sin  ^. 
i(log3-i). 


V/     20.    j- 

f  ^JC 

J      21.  , 

^  J  ^  I   —   COS  X 


tan  \x. 
I  —  cot  \x. 


^  -■  It 


dx 


±  sin  jc ' 
Multiply  both  terms  of  the  fraction  ^  i  =F  sin  x.  tan  x  ±  sec  jp. 


vi    ^*^*    JsecQ  ±  tan 9' 

log  tan    -  +  - 

L4       2_ 

±  log  cos  9. 

Nl   24.     cos  0  cos  39  ^9.  ■  See  Art,  33. 

J  sin  4S  +  i  sin  29. 

IT 

\/  25.      ^  cos  9  COS  29  rtTs, 
•'0 

I 
3* 

J   26.    rsin'9  sin  20 //9, 

It 

isin*9l'=i 
—^0 

\/  27.      ^  sin  39  sin  29  d% 

' 

2 
5' 

46  ELEMENTARY  METHODS   OF  INTEGRATION.     [Ex.  III. 


\\      28.       sin  m{}  cos  «0  ^0, 


I  —  cos  (m  -\-  n)e       I  —  cos  (m  —  n)  B 
2  {m  -\-  n)  2  {m  —  n) 


I     29.      cos  X  cos  2x  cos  3jr  ^JC, 

Reduce  products  to  sums  by  means  of  equation  (2),  Art.  33. 

I  Fsin  Q>x      sin  aa:       sin  2x         "1 
4L     6  4  2  J 

|/ (l  —  COS:r)^;xr,  2  V'2. 

1     31.     ^ -i ^  ri    '  -2      >  —  tan-M  -tan^    . 

\1     "^      }  a  cos^  X  +  i?  sm  X  ab  \_a  J 

\  f        ^jt:  I  ,  tannic 

4      ^2.     i— ,  -r- tan" 

^     -^       J I  +  cos'  X  ' 


"'Z   2^-    J d'  cos'  ^  -  ^'  sin' x'  2ab  ^^^ 


4/2  4/2 

d;  +  ^  tan  9 


sin  X  dx 


V  (3  cos'  a:  4-  4  sin'  ^) 
sin  x  cos'  ^  dJ^c 


2  «/^     ^  a  —  b  tan  Q 
COS"*  J^  cos^^. 


,  f  sm  jg  COS  X  di 

>4     35-   J  J  _j_  ^2  CQs'^  X 

r  y^ 

Putting  y  /<?r  cos  x,  //^^  integral  becomes  —    — — — j-^ 


COS  ^      tan  *  {a  cos  :r) 
a  a 


§IV.] 


EXAMPLES. 


47 


36.     f— T 


dB 


b  smQ 
/'a/  sin  6  =  cos  (6  —  |-7r),  ^«^  use  formulas  (G)  and  (6^). 

2  ,  r    /^  —  /^         2Q  —  7r~| 

V  (^  +  «)  +  4/  (<^  —  «)  tan  (i-  0  —  i  Tt) 


\ia<b, 


( ..  J- 


V  (b'  -  a')  ^^^  Vib  +  a)-  V  {b  -  a)  tan  (iQ  -  i  n) 


dQ 


3  +  5  cos 


ilog 


2  +  tan^Q 


^^>-  b 


\S  40.  J 


5  +  3  COS  0 
do 


cos  0 


2   COS  G  —  I 


2  —  tan  f  B 
i  tan"* [Han  J 6]. 

|-tan-*i3tan|Q{. 


J_wL-^^3_tani^ 

♦^3      ^  I  +  4/3  tan  i  0  • 


3  —  cos 


tan"*  V2 


COS  9 
^0 


2  4/3* 


^  cosOj' 


6V^  Arl.  36. 


-;  cos 


,  e  +  cos  9 


sm  9 


^-  rfTT 


(i  _  ^2^^  ^  +  ^cosQ       I  —  r  I  +  ^cos9 

(2  +  /)  TT 


(i  +  ^cos9)' 


2(1 -.')i 


48  ELEMENTARY  METHODS  OF  INTEGRATION.  [Ex.  III.. 


f  p  COS  X  -\-  q^WiX  . 

45-    \    7\-' — dx, 

^     J  a  cos  :v  +  /?  sm  jp 


Solution : — 

By  adding  and  subtracting  an  undetermined  constant,  the  fraction 
may  be  written  in  the  form 

p  cos  X  +  ff  sin  X  -h  A  (a  cos  ^  +  /^  sin  ^) 

rj~- — —  ~  ^» 

a  cos  :<[:  +  /?  sm  jc 
we  may  now  assume 
/  cos  X  -^  q  sin  X  +  A  {a  cos  :r  +  ^  sin  .^t:)  =  ^  (<^  cos  x  —  a  sin  x) ; 

the  expression  is  then  readily  integrated,  and  A  and  k  so  determined 
as  to  make  the  equation  last  written  an  identity.     The  result  is 

f /» cos  ^  +  ^r  sin  Jic   ,         ap -^  bq  bp  —  aq .       .  ,    ?  \ 

— ^^ —  dx  =  ^.   .    ,o  X  +     3  ,    ,■;  log  (a  cos  ^  +  <^  sm  x). 

J  a  cos  X  -\-  bsmx  a^  ^-  b'  a^  +  b'      °  ^  ' 

46.     — ; — T— ,     See  Ex.  At^. 

ax       ^         b       .       f  ,    7    ■      \ 

rr>  +     a     I     z2  log  (a;  cos  ^  +  ^  Sm  X). 


a'  +  b'   '   a'  -]-  b' 

47.  Find  the  area  of  the  ellipse 

X  =  a  cos  ^  ji^  =  (^  sin  ^. 


—  4ab\ 


o 

sin^  dfdS  =  nab. 


48.  Find  the  area  of  the  cycloid 

jp  =  d!  (^  —  sin  ^)  jF  =  <^  (i  —  cos  ^). 


(27r 
(i  -  COS  tpY  dip  =  3a'7r. 
o 


§111.]  EXAMPLES.  49 

49.  Find  the  area  of  the  trochoid       (b  <  a) 

X  =  aip  —  dsintp  y  =  a  —  d  cos  ^'. 

50.  Find  the  area  of  the  loop,  and  also  the  area  between  the  curve 
and  the  asymptote,  in  the  case  of  the  strophoid  whose  polar  equation  is 

r  =^  a  (sec  0  ±  tan  0). 
Solution  : — 
Using  0  as  an  auxiliary  variable,  we  have 

/     .     •     \                            n          ,   sin'^Q"! 
ji-  =  ^  (i  ±  sin G)  y  =  «    tan  Q  ± L 

^  '  "^  L  cos  Qj 

the  upper  sign  corresponding  to  the  infinite  branch,  and  the  lower  to 
the  loop.     Hence,  for  the  half  areas  we  obtain 

+  a"  [  "sin  BdB  -^  a'[    sin'  G  ^0  =  ^H  i  +  - 

and  — «'  I    sin  B  dB  +  d^  f    sin'  B  dB  =  a^\  i . 

Ji.  )^  L.         4J  . 


50  METHODS   OF  INTEGRATION,  [Art.  37. 

CHAPTER   II. 

Methods  of  Integration — Continued. 


IV. 

Integration  by  Change  of  Independent  Variable, 

37.  If  X  is  the  independent  variable  used  in  expressing  an 
integral,  and  y  is  any  function  of  x,  the  integral  may  be  ex- 
pressed in  terms  of  j,  by  substituting  for  x  and  dx  their  values 
in  terms  of  y  and  dy.  By  properly  assuming  the  function  j, 
the  integral  may  frequently  be  made  to  take  a  directly  integra- 
ble  form.     For  example,  the  integral 

[     X  dx 


J  {ax  +  bf 
will  obviously  be  simplified  by  assuming 

y  —  ax  +  b 
for  the  new  independent  variable.     This  assumption  gives 


X  =  - ,                       whence                    dx  —  — 

a                                                                      a 

substituting,  we  have 

X  dx           I 
J  {ax  +  b)^  ~  a"  J 

\{y-b)dy 
f 

I 

iog7  +  ^^; 

§  IV.]  CHANGE   OF  INDEPENDENT    VARIABLE.  5 1 

or  replacing/  by  x  in  the  result, 

[     X  dx  I  1       /        .    A\   ,  ^ 

38.  Again,  if  in  the  integral 

f    dx 

Jf'-  I 
we  put  y  —  e%     whence 

X  =  log  7,  and  dx  =  —  , 

we  have 

f    dx    _  r       dy 
Jf"—  I  "~  J  j(j—  I)* 

Hence,  by  formula  (A),  Art.  17, 

It  is  easily  seen  that,  by  this  change  of  independent  variable, 
any  integral  in  which  the  coefficient  of  dx  is  a  rational  func- 
tion of  £%  may  be  transformed  into  one  in  which  the  coefficient 
of  ^  is  a  rational  function  of  7. 

Transformation  of  Trigonoinetric  Forms. 

39.  When  in  a  trigonometric  integral  the  coefficient  of  dB  is 
a  rational  function  of  tan  ^,  the  integral  will  take  a  rational 
algebraic  form  if  we  put 

dx 
tan  6  =  Xy  whence  dd  = 


i+;r2 


52  METHODS   OF  INTEGRATION.  [Art.  39. 

For  example,  by  this  transformation,  we  have 

f       de        _  f  dx 

Ji  -4-  tan6'~  J(i  +;t^)(i  +;ir)* 

Decomposing  the  fraction  in  the  latter  integral,  we  have 

f        ^^        _   \{    dx  If  X  dx     ^\{    dx 

J  I  -h  tan  d~~  2)1  -^  x^       2  J I  +  x^     '   2JI  +  X 

=  ^  tan"\'  —  i  log  (i  +  x^)  4-  i  log(i  +  x) 

'''■         \i  +  tan  ^  =  i  C^  +  ^^S  (^°'  ^  +  ^^"  ^)3- 

40.  The  method  given  in  the  preceding  article  may  be  em- 
ployed when  the  coefficient  of  d6  is  a  /lomogeneous  rational  func- 
tion of  sin  Q  and  cos  6,  of  a  degree  indicated  by  an  even  integer ; 
for  such  a  function  is  a  rational  function  of  tan  Q.  It  may  also 
be  noticed  that,  when  the  coefficient  of  dd  is  any  rational  func- 
tion of  sin  S  and  cos  l9,  the  integral  becomes  rational  and  alge- 
braic if  we  put 

e 


for  this  gives 

2Z 


sin  e 


I  + 


c^' 


^  =:=  tan  ^  ; 

cos  ^  =  ^^^3, 

This  transformation  has   in  fact  been  already  employed  in 

the  integration  of .     See  Art.  x^, 

^  a  +  b  Qos  6  ^^    . 


§  IV.]       LIMITS  OF    THE    TRANSFORMED   INTEGRAL.  53 

Limits  of  the   Transformed  Integral. 

41.  When  a  definite  integral  is  transformed  by  a  change  of 
independent  variable,  it  is  necessary  to  make  a  corresponding 
change  in  the  Hmits.     If,  for  example,  in  the  integral 

r       dx 

we  put  X  —  a  tan  ^,  whence  dx  —  a  sec^O  dO, 

we  must  at  the  same  time  replace  the  limits  a  and  oo ,  which 
are  values  of  x,  by  ^Tt  and^;r,  the  corresponding  values  of  d. 
Thus 


L(^F^.=,-J-I:cos^^ 


dd 


~  2a^L 


6  -h  sin  6  cos  6 


7t  —  2 


The    Reciprocal  of  x  taken   as    the   New  Independent 

Variable. 

42.  In  the  case  of  fractional  integrals,  it  is  sometimes  use- 
ful to  take  the  reciprocal  of  x  as  the  new  independent  variable. 
For  example,  let  the  given  integral  be 


dx 


}x^{x  -h  if 
Putting  ^  =  -y  whence  dx  = ^ , 

y  y 


54  METHODS  OF  INTEGRATION.  [Art.  42. 


we  have 


y 

\      y. 

Transforming  again  by  putting  z  —  y  ■\-  I,  the  integral  be- 
comes 


=  _  _  +  3^  _  3  log  ^  -  - 


Therefore,  since  ^  =  r  +  i  =  -  +  i  = 

X  X 


dx_     _  _  (x^  \f      3(-^'+  0 f .  1  „^  X  -^  I 

(,r  +  1)2  ~  2x^  X  X  ■\-  \       ^      ^      X 


A  Power  of  x  taken  as  the  New  Independent  Variable. 

4-3.  The  transformation  of  an  integral  by  the  assumption, 

y^x'^ (i) 

is  not  generally  useful,  since  the  substitution 

-  I    -- 1 

X  =  r«,  whence  dx  —  -  y""      dy, 

11 

will  usually   introduce  radicals.     Exceptional  cases,  however, 


§  IV.]  THE  EMPLOYMENT  OF  POWERS  OF  X.  55 

occur.     For,  since  logarithmic  differentiation  of  equation  (i) 
gives 

—  -=  —  , (2) 

X       ny 

it  is  evident  that,  if  the  expression  to  be  integrated  is  the  product 
of  —  and  a  function  of  x^ ,  the  transformed  expression  will  be 

the  product  of  —  and  the  like  function  of  y. 
For  example,  the  expression 

{x!"  -  i)  dx 
x{x^  ^  I)  ' 

dx 
which  is  the  product  of —  and  a  rational  function  of  ;r*,  becomes 

dy , 


Ay{y^  I) 


a  rational  function  of  y.     Hence,  decomposing  the  fraction  in 
the  latter  expression,  we  have 

J  ^(^+  I)      a)  yiy^i)  ^      4    ^       y 


V  (x^  +  i) 


44.  When  this  method  is  applied  to  an  integral  whose  form 
at  the  same  time  suggests  the  employment  of  the  reciprocal, 
as  in  Art.  42,  we  may  at  once  assume  y  =  x~'^,  ^  Thus,  given 
the  integral 

["        dx 


56  METHODS  OF  INTEGRATION.  [Art.  44. 


putting                y  =  x^, 

whence 

dx  _       dy 

~^  ~  "  sy' 

we  obtain 

- 

_ir  ydy 
3J12J/+  I 

=  - 

y  ,  log  {2y  + 
6"^           12 

i)^ 

°_2-log3 

12 

45.  The  same  mode  of  transforming  may  be  employed  to 

dx 
simplify  the  coefficient  of  —  ,  when  this  coefficient  is   not  a 

rational  function  of  x^.     Thus,  the  integral 

r        dx 

}xV{x^-a^) 

will  take  the  form  of  the  fundamental  integral  (/'),  if  we  put 

09  .  dx       2  dy 

x^  —  Ti  whence  — —  _.^i-. 

X       I  y 
Making  the  substitutions,  we  have 

dx  2[  dy  2  -X  y  2  _i  (x\  5 

—     '  -^  —         sec     -4-  =  — -,  sec      '      ^ 


f     ^^      —  ?  f      ^y 

]xV(^-ci^)  "  3J yV{7~^~^)  "  ^J  ^^^      J  "  3^1  \a 


Examples  IV. 


^  I-  j  l^^^'^'>  log  (2  +  '^')  +  ^ 


I 


§iv.] 


EXAMPLES. 


57 


/  . 

'     xdx 

J(i-^r' 

^3. 

f^'--^+'^r 

J(2^  +  ir    ' 

v/4. 

r    .v'<fe 

J-xGr+2)" 

/s. 

r    ^^ 

J  I  +  f-^ 

' 

6. 

f     ^ 

e-r  _  ^-x  > 

p           £2^  ^ 

7- 

J  -co  f-  +   I   ' 

8. 

f  '"+'    ^X 

Jl-6-'^-^' 

'  2  +  tan  6 

9- 

.  3  —  tan  e     ' 

r     ^6 

J  tan'  6  -  i' 

'  tan'  0  ^0 

tan'  6  —  I ' 

cos  0  d^ 

a  cos  6  —  ^  sin  Q ' 

2^—1 

2(1-  xf- 

2A  -f    I    _    log  \2X  +1)  7 

8  "1  8(2;*:+  i) 


log;/  + 


4y  —  2 


]-log.-i, 


^—  (log  I  -f  f-*^). 


2    °^  f ^  +  I 


I  —  log  2, 


€-»^  +  2  log  (f-*  —  l). 


e  —  log  (3  COS  0  —  sin  e) 


I  ,      tan  0—1       0 

—log 

4     °  tan  0  +  1      2 


I  ,      tan  0  —  1,0 

-  log +  - , 

4     ^  tan  0  +  I       2 


aB  —  b  log  (a  cos  0  —  <^  sin  0) 
a'  +  b' 


58  METHODS  OF  INTEGRATION,  [Ex.  IV. 


f      COS  6  ^&  r»    ^  .' 

•^    Jcos(a'  +  ''^' 


(9  +  oc)  cos  «  —  sin  « log  cos  (9  +  a). 


[  sin  (0  '+  a) 

(6  +  /5)  cos  {a—  ft)  +  sin  (or  —  /?)  log  sin  (9  +  )5). 


5.      tan  (9  +  «:)  cos  9  ^/9,    —cos  9  +  sin  <^  log  tan 


2B   +  2a  -\-  7t 


,     {'^     COS  B  dB  ,       .  .    ,         . 

'   Jo  sin  (a:  +  9)^  cos  «  log  (2  cos  Of)  +  Of  sm^. 

IT  IT 

rr  cos  j  9  ^^  I    ^       1^2+2  sin9n6  _  log  (3  +  2  ^2) 

Jo    COS  9        '  4/2     °  ^2  —  2  sin  9  Jo  4^2 

_     fsin|9^/9  ,      ^ 

18.     ^^-r— ,  log  tan 

J     sin9  ^ 


TT  + 


fx'  dx  /z' 


20. 


l^'(l+-v')'  '°^ 

r*    jtr'^jT  I  fs 

Jo  (l    +  -^T'  4  Jo 


4/(1  +  x")        _I_^ 
a;  2ar'' 


21.    I    7-— T-— iTs,  -  I  ~  sin"  2B  dB—  —r 


§  IV.]  EXAMPLES,  59 


n 


Jdx  L  4-  £  _  1 


X  -\-  \ 

X 


[        dx  I  I        ,   ,  X 

^^-   Jx(^'-i)'  flog(^'-.)-logx. 


o 

_  2    - 

-log  3 

I 

8 

I 

;'°s^ 

x' 

4^ 

+  ^.rV* 

I 

Inor  — 

Jt:^ 

^n      &^«  +  ^a' 


e/jf 


2  _       /^ 


V. 

Integrals  Containing  Radicals, 

46.  An  integral  containing  a  single  radical,  in  which  the 
expression  under  the  radical  sign  is  of  the  first  degree,  is 
rationalized,  that  is,  transformed  into  a  rational  integral,  by- 
taking  the  radical  as  the  value  of  the  new  independent  vari- 
able.    Thus,  given  the  integral 

f  dx 


14-  v{x-\-\y 


60  METHODS  OF  INTEGRATION,  [Art.  46. 


putting 

J  =  V(^  +  I), 

whence 

X  —  f  —  I,                   and                   dx  =  27  cfy\ 

we  have 

f           ^- 

J  I  +   4/(;r  + 

-Ay^y  =  2\dy  2f  '^y 

i)         Ji  +7         J  "^         Ji  +j/ 

=  2j-2log(l   +/) 

=  2^/(x  +  l)  -  2log[l  +  »/{x  +  l)]. 

47.  The  same  method  evidently  applies  whenever  all  the 
radicals  which  occur  in  the  integral  are  powers  of  a  single 
radical,  in  which  the  expression  under  the  radical  sign  is  linear. 
Thus,  in  the  integral 


dx 


,(^-_i)5  +  (^--i) 


±  » 


the  radicals  are  powers  of  (;ir  —  i)^ ;  hence  we  put  y  =  {x  —  i)K 
and  obtain 


dx  ^^f^    fdy 


=  6\\y-  i)dy  +  6f  -f^=  -3  +6\og2, 

Jo  Jo  J  +    I 

48.  An  integral  in  which  a  binomial  expression  occurs 
under  the  radical  sign  can  sometimes  be  reduced  to  the  form 
considered  above  by  the  method  of  Art.  43.  For  example, 
since 

f        dx 
ix{x^-^  i)^ 


§  v.]  INTEGRALS  CONTAINING  RADICALS.  6l 

fulfils  the  condition  given  in  Art.  43,  when  n  —  3,  the  quantity 
under  the  radical  sign  may  be  reduced  to  the  first  degree. 
Hence,  in  accordance  with  Art.  46,  we  may  take  the  radical  as 
the  value  of  the  new  independent  variable.     Thus,  putting 


whence  ^  =^  ^  —  i,  and 

we  have 


dx        4js'^  dz 


X  -3(^-1)' 


{       dx         _4  fj^ dz 
ixix^  +  i)i~3  J^-i* 

Decomposing  the  fr^^ction  in  the  latter  integral  as  in  Art.  20, 
we  have  finally 

_^  —  _  tan'M  (;i^  +  I)      +  -  log^- \ — -  . 


Radicals  of  the  Form  V{ax^  +  3). 

49.  It  is  evident  that  the  method  given  in  the  preceding 
article  is  applicable  to  all  integrals  of  the  general  form. 

\x'^'''-^'{ax^  i-  dy+^dx, (I) 

in  which  m  and  7i  are  positive  or  negative   integers.     These 
integrals  are  therefore  rationalized  by  putting 

y  =  V[a^  +  b).. 


62  METHODS  OF  INTEGRATION,  [Art.  49. 


Putting  m  =  O,  the  form  (i)  includes  the  directly  integrable 
case 


f(^,i^  +  by 


+  *  xdx. 


50.  As  an  illustration  let  us  take  the  integral 

dx 


f.7 


X  V{x^  +  a^)  ' 
putting  J  r=  V{^  +  a^), 

,  2      5      2  1  ^^       y  ^y 

whence  x^=t  —  ^  ,  and  —  =  ■  /    ^  , , 

X      y  —  a'^ 

we  have 

dx  _  r     </j/ 

Hence,  by  equation  (^')  Art.  17, 


dx         _  i^       y  ~  ^  _  ^  \      ^(^'^  +  ^^)  —  ^ 


Rationalizing  the  denominator  of   the  fraction  in  this  result, 
we  have 

V{x^  -^  d')-a  ^  r  1  (-^-^  4-  a')  -  df 
Vix^  +  a^)  -{-  a~  x^ 


Therefore 


g  v.]  INTEGRALS  CONTAINING  RADICALS.  63 

In  a  similar  manner  we  may  prove  that 


51.  Integrals  of  the  form 

\x^"'{ax^  -VbY^^dx (2) 


are  reducible  to  the   form  (i)  Art.  49,  by  first  putting  j  =  -. 

X 

For  example : 

t       dx 


{ax^  +  bf 


is  of  the  form  (2) ;  but,  putting  x  —  -  ,  whence 


^(^^  +  ^):.i^(fL±i>3  and  dx  =  -%, 

we  obtain 

f        dx        _       r      y  dy 
J  (ax"  +  bf  ~  ~  J  '{a-\-bff 

The  resulting  expression  is  in  this  case  directly  integrable. 
Thus 

[         ^^  ^  ^  _  ^  /<vx 

\a^-Vb\^      bV{a-^bf)      b^{a^ -^r  h)'     •     •     U; 


64  METHODS  OF  INTEGRATION.  [Art.  52. 


Integration  of 


52.  If  we  assume  a  new  variable  s  connected  with  x  by  the 
relation 

z-x=  t/(.t^±^), (I) 

we  have,  by  squaring, 

^  —  2SX  =  ±  a^, (2) 

and,  by  differentiating  this  equation, 

2{2  —  x)  dz  —  2z dx  =Q\ 
whence 


dx 

dz 

Z  —  X 

~  z' 

dx 

(3) 


or  by  equation  (i), 

X         _  dz 

Integrating  equation  (3),  we  obtain 

63.  Since  the  value  of  x  in  terms  of  z^  derived  from  equa- 
tion (2)  of  the  preceding  article,  is  rational,  it  is  obvious  that 
this  transformation  may  be  employed  to  rationalize  any  ex- 

dx 
pression    which    consists   of   the  product  of—^^^ — -g.    and  a 

rational  function  of  x.     For  example,  let  us  find  the  value  of 


\^V{^±a^)dx, 


§  v.]  TRIGONOMETRIC   TRANSFORMATION.  65 


which  may  be  written  in  the  form  -w^  ^ 

dx 


\{:^±a^ ^^-  ^V'      r    A 


By  equation  (2) 


2Z 

whence 


S 


^±^=(^! 

4^ 
Therefore,  by  equations  (3)  and  (5), 

\^V{x'±d^dx  =  -^^^-^^dz 

If     ,     ^  €?  [dz  ^  a^  [dz  ^^  ^' 

4J  2  J^       4  J-s*  1^       "T^ 

By  equations  (4)  and  (5),  the  first  term  of  the  last  member 
is  equal  to  J  jt  V{^  ±  d^).     Hence 

[Vix'  ±  d^)dx  =  ^^^'^  ^  ^  ±  -  log  [x  +  V{x'±d')}  .    .    (Z) 

Transformation  to    Trigonometric  Formes. 
54.  Integrals  involving  either  of  the  radicals 
^(c^-x"),  Vid'  +  x^),  or  V(^-^ 


^^  METHODS  OF  INTEGRATION,  [Art.  54: 

can  be  transformed  into  rational  trigonometric  integrals.     The 
transformation  is  effected  in  the  first  case  by  putting 

X  ^  a  sin  ^,  whence  *^{c^  —  x^^  =  a  cos  6 ; 

in  the  second  case,  by  putting 

X  =  a  tan  ^,  whence  \/{c?  -\-  x^")  =  a  sec  6  ; 

and  in  the  third  case,  by  putting 

X  =  a  sec  6^,  whence  V{^  —  c?)  —  a  tan  6, 

55-  As  an  illustration,  let  us  take  the  integral 
f  i/(^2  _  ^2^  ^^  . 

putting X  =  asm  6,  we  have  V{a^.  —  x^)  =  a  cos  6,dx  =  a  cos  6  dd\ 
hence 


{v{^-^)dx=a^{cos^e 


dS 


(T  6      c?  sin  6  cos  6 

=  ^+ — 5 — ' 

by  formula  (Q  Art.  29.     Replacing  (9  by  x  in  the  result, 

Regarding  the    radical    as   a  positive   quantity,  the  value 
of    ^  may  be   restricted  to  the  primary  value  of  the  symbol 

sin  - '  —  (see  Diff.  Calc,  Art.  54)  ;  that  is,  as  x  passes  from  —  a 

to  +  ^,  ^  passes  from  —  \n  \.q  +  J  tt. 


§  v.]  INTEGRALS  CONTAINING  RADICALS.  67 


J 


Radicals  of  the  Form  ^(a:x?  +  bx  +  c). 


56.  When  a  radical  of  the  form  V(a^  +  bx  +  c)  occurs  in  an 
integral,  a  simple  change  of  independent  variable  will  cause  the 
radical  to  assume  one  of  the  forms  considered  in  the  preceding 
articles.     Thus,  if  the  coefficient  of  x^  is  positive, 

in  which,  if  we  put;r4-  — =jK,    the   radical   takes   the    form 

V{y^  +  a^)  or  V{y^_  —  cF)-,  according  as  /i^c  —  b^  is  positive  or 
negative.  If  ^  is  negative,  the  radical  can  in  like  manner  be 
reduced  to  the  form  ^/{c^  —  f)  or  |/(—  a^—y^)  ;  but  the  latter  will 
never  occur,  since  it  is  imaginary  for  all  values  of  j/,  and  there- 
fore imaginary  for  all  values  of  x. 

For  example,  by  this  transformation,  the  integral 

f  dx 


J  {a^  ^bx  +  f)t 
can  be  reduced  at  once  to  the  form  {J),  Art.  51.     Thus 
dx  f  dx 


fax  f  u^ 


2a  A^x  +  20 

4^ 


68  METHODS  OF  INTEGRATION.  [Art.  57. 

57.  When  the  form  of  the  integral  suggests  a  further 
change  of  independent  variable,  we  may  at  once  assume  the 
expression  for  the  new  variable  in  the  required  form.  For 
example,  given  the  integral 

V{2ax  —  x^^  X  dx\ 

we  have  i/(2ax  —  x^)  =  \/[a^  —  (^  —  a)^] 

hence  (see  Art.  54),  if  we  put  x  —  a  =  a  sin  6,  we  have 
V{2ax  —  x^)  =  a  cos  ^, 
;ir  =  ^  (i  +  sin  ^),  dx=  a  cos  6  dO  ; 

,-.|  V{2ax  -  x'')xdx  =  c^  [cos2  d{i  +  sin  d)  dd 

=  ^{d-\-sme  cos  8)-  —  cos8  6 

c^    .       X  —  a      a ,  .    ,,  ox       I,  «a. 

=  —  sm-^ +  _(^  _  ^)  ^(2ax  -x^)  —  -  (2ax  -  x^y 

=  —  sin  - ^ +  ^  V{2ax  —  x^)  \2x^  —  ax  —  3^]. 

The  Integrals 

f dx^ ,     f       dx 

JV[(^--)(^-^)]  JV[(^-«)(/^-^)]- 

58.  An  integral  of  the  form  J  —r— ^ — y ^  may  by  the 

method  of  Art.  56,  be  reduced  to  the  form  {K\  Art.  52,  or  to 
the  form  (7'),  Art.  10,  according  as  a  is  positive  or  negative. 


§  v.]  IRRA  TIONAL  INTEGRALS.  69 

But  when  the  quantity  under  the  radical  sign  can  be  resolved 
into  linear  factors,  the  formulas  deduced  below  give  the  value 
of  the  integral  in  forms  which  are  sometimes  more  convenient. 
If  a  and  ^  are  the  roots  of  the  equation 

ax^  -V  bx  ^  c  —  Oy 
the  integral  may  be  put  in  the  form 

dx 


I    f  dx^ I        f 

Ta]^\{x~aMx-b)\     ""'     VT^^)]' 


Va  J  ^/\{x  - a){x  -  fS)\  ^/{-d)]  V[(x  -  a){p - x)]  ' 

according  as  a  is  positive  or  negative.     Assuming 

V{x  —  a)  =2,       whence        x  =  j^  -h  a         and         dx  =  2zdz^ 

we  have 

by  formula  (K),  Art.  52  ;  hence 

In  like  manner  we  have 

f dx^ f  dz  _       .  _  z 

J  V\_{x-a){fS  -x)-]-^  ]^^-a-^)  ~  ^  "'""    V{^  -  a)  ' 

by  formula  (/') ;  hence 

f  dx  .      ^  ./  X  —  a  I  ^^ 


70  METHODS  OF  INTEGRATION.  [Art.  58. 

It  can  be  shown  that  the  values  given  in  formulas  (TV)  and 
{O)  differ  only  by  constants  from  the  results  derived  by  em- 
ploying the  process  given  in  Art.  56. 


Examples  V. 

V    I.      ^(a  —  x)'X  dx^  [a  —  x)^  (3^  +  2d), 

c  ^ 

2.       V{^  +  d)'^^  dx^     -  («  +  a:) 2  _  1-  (^  +  x)\  H {a  -\-  x)\, 

(X  dx  2     3 

^-^-^,  -x^  -  .r  +  2  ^^  -  2  log  (i  +  ^x). 

5-  J  ^^"1  j>  21/^  +  2  log (i  -  Vx), 

J-a^                ^  7                 4     Jo                   2d> 

(dx  2        _      /2^  —  a 

— T/ 2\  >  -  tan  '  1/ - 

jc  V(2ax  —  a)  a             ^        a 

Jo^  9          7           5    Jy-      315 

^          J2:xr?  — ^*  4          4                    8 


v.]  EXAMPLES.  71 

.10.    r(.+  x)t.^.,  ^'_3/-|^=^3^^^ 

Jo  o        5  Ji  10        40 

^^-    1^(7-%  ^-^^  2  (1+  ^)  V(l  -  at).     _ 

Jxdx  T,     •       ,. 
-7^ r\ »                  Rattonahze  the  denominator. 


X'  +  f^'*  -  a'Y 


Za^ 


/  f dx^ 2  (.y  +  g)«  -  2  (^  +  3)^ 

V      13.   J  ^(^  +  ^)  +  ^(^  +  ^) »  3  («  -  ^) 

/  [V{x^-VT)dx  V{x*  +  ^)    ,    ^  Inr  ^^^^'  +0-1 

;  ^'  J        ^        '  2       +4^°^  v(;t;^4-I)^-I• 

"J.(?+^'    J-"'-?],  'b"^'- 

X 

«^{x^  — c^\  —  CL  sec  ~ '  -  . 

etc 


72  METHODS  OF  INTEGRATION.  [Ex.  V. 


r      x'dx 

'    ~       ^/  .a    , — ^d^\     See  formulas  (Z)  and  {K). 


jXi/(x'  -i-a')--  a' log  [x  +  i/^  +  a')] 


20.     — ^^ ^ dx^  a  log ^^ +  4/(a   —  A^  ). 


—,  V(^r'  +  «')  +  -log  \x  +  4/(-a:'  +  «')]  -  ^ 

2^  2  2^ 


log  [  |/^:v'  +  a')  +  .rj ^ ^ ^ . 

\  X 

I  C      X  dx  T 


§V.] 


EXAMPLES, 


73 


25.      ^{ax"  +  b)  dx,  [a  >o]         Put  V(ax^  ■{■  b)  =  z  —x  i/a. 

-^  log  \_x  s/a  +  ^(ax"  +  ^)]  +  -  ^  '^(ax^  ^b). 

j  2  Va  2  - 


I        Xq  ^  +  '*^(-^'  +  ^')  4-  «  -  Via"  +  ^') 


V{d'+b')  ^x  +  V{^"  +  b')-^  a  +  V{d'  +  b') 


27 


f         dx 


Vii  +x') 


^^'  \ixWii-x')  ' 


G; 


cot  H    =  Vs- 


29 


V{^'-a') 


i dx 

^V(^'-i)' 


^^'    Jo  (>  +  «'] 


I 

2 

P 

2A;'                   2 

log  tan  \        \  . 

74  METHODS  OF  INTEGRATION.  [Ex.  V. 


ZZ- 


3^ 


(dx  I  J        ::c4/2 

^   '   Jo  4/(^-^0   L      Jo  V(cix-x')_\ 

2 

f  ^.V 

38.  3,4 V  ^^^^ 

39.  Y(2(ix  ~  x^)'dx, 

40.  )/(2ax  —  x^)-x  dx, 

a^      ^  cos''  0  (i  4-  sin  6)  .'/Q  =  d;' . 

2 

41.  ^{2ax  —  x'^)'X^  dx, 

a'  j°  ^  cos'  0  (i  4-  sine)'  ^0  =  a'  p^  -  -~] . 


^  =  2r  =  sec 


I  4  sine)VQ  — 

T6" 

—  X 

X 

2X^ 

dn 

4 

§  v.]  EXAMPLES,  75 


j- dx_ 

J  ^y2ax  +  x^) 


42.  ./ ,      .,:iv> 


by  Art.  56,         log  \x  -\-  a  ^  */(2ax  +  .r')]  +  C; 
^j;  ^r/.  58,  log  [  »^x  -r  i/(2d;  +  a)]  +  C". 

,J{2ax  +  .r')  '   ■^^'"'"^"  "^  ■^'^  ""^  '°^  [-t-  +  «  +  V(2ax  +  ^■•')]. 

,,.  [/_z_^.vr=[ ' ---^-^-  .1, 

^^    J  '^   2^  —  .r        |_      J  1/(2^20:  —  .r')  J 


tf  sin- V{2ax  —  Jt-'). 


45.  j  ^(3+^l^_^^) '  ^i'  ^-^-  56,  sin-  ^  :^^  +  C; 


^^  Art.  58,  2  sin-^  ^  "^4^  +  ^'• 


46.       -77 3T,  2sin-H/-       =.-;r. 

47-  JV(3  +  ,^_^')'3 

49-  £V^'^'  logCs  +  ^l'^ 


sin- .  ^~^  -  (-^  +  3)  ^(3  +  2-^  -  ■^'') 
2  2 


"7^  METHODS  OF  INTEGRATION.  [Ex.  V, 

J __- 

^      50.  Find  the  area  included  by  the  rectangular  hyperbola 

y^  =  2ax  +  x^^ 
and  the  double  ordinate  of  the  point  for  which  x  =  2a. 

al6V2  —  log  (3  +  2  V2)]. 

Find  the  area  included  between  the  cissoid 

X  {x^  +  /)  =  2ay^ 


I  51.  Fi 


and  the  coordinates  of  the  point  {a,  a)  ;  also  the  whole  area  between 
the  curve  and  its  asymptote. 


J 


(—  7t  —  2  W^,        and         ^na^. 


52.  Find  the  area  of  the  loop  of  the  strophoid 

x{x'  +/)  +  a{x'  -f)  =  o; 
also  the  area  between  the  curve  and  its  asymptote. 

,       and        20^  ii  -^ j 


/jr    -U    jQ 

For  the  loop  put  y  z=z  —  x        ^ -j-  ,  since  x  is  negative  between  the  limits 

—  a  and  o. 
yj        53.  Show  that  the  area  of  the  segment  of  an  ellipse  between  the 

X 

minor  axis  and  any  double  ordinate  is  ab  ^\Vi.-^  — V  xy. 


§  VI.(  INTEGRATION  BY  PARTS,  77 

VI. 

Integration  by  Parts.  __ 

59.  Let  u  and  v  be  any  two  functions  of  x ;  then  since 
d  (uv)  =  udv  +  V  duy 

udv  -i-lv  dUy 


uv 


whence 


\udv=uv—\vdu (i) 


By  means  of  this  formula,  the  integration  of  an  expression 
of  the  form  udVy  in  which  dv  is  the  differential  of  a  known 
function  v^  may  be  made  to  depend  upon  the  integration  of 
the  expression  v  du.     For  example,  if 


u  - 
we  have 

=  cos-';ir 

and 

I 

iv  =  dxy 

du  = 

dx 

hence,  by  equatioi 

Vii- 

^y 

COS" 

'  X'dx  = 

;r  cos-^;i: 

+   - 

xdx 

'(I  -A 

in  which  the  new  integral  is  directly  integrable ;  therefore 

cos-^;r-^;ir  =  x  Q.o?>~^ x  ^  4/(1  —  x^'). 
The  employment  of  this  formula  is  called  integration  by  parts. 


7.8 


METHODS  OF  INTEGRATION. 


[Art.  60. 


Geometrical  Illustration, 


60.  The  formula  for  integration  by  parts  may  be  geomet- 
rically illustrated  as  follows.  Assum- 
ing rectangular  axes,  let  the  curve  be 
constructed  in  which  the  abscissa  and 
ordinate  of  each  point  are  correspond- 
ing values  of  v  and  u^  and  let  this 
curve  cut  one  of  the  axes  in  B.  From 
any  point  P  of  this  curve  draw  PR 
and  PS^  perpendicular  to  the  axes. 
Now  the  area  PBOR  is  a  value  of  the 

indefinite  integral    u  dv,   and    in    like 
manner  the  area  PBS  is  a  value  of  \vdu', 
and  we  have 


Area  PBOR  =  Rectangle  PSOR  -  Avesi PBS; 


therefore 


ludv  =  uv  —  \v  du. 


Applications, 

61.  In  general  there  will  be  more  than  one  possible  method 
of  selecting  the  factors  u  and  dv.  The  latter  of  course  in- 
cludes the  factor  dx,  but  it  will  generally  be  advisable  to  in- 
clude in  it  any  other  factors  which  permit  the  direct  integra- 
tion of  dv.  After  selecting  the  factors,  it  will  be  found  con- 
venient at  once  to  write  the  product  u-v,  separating  the  factors 
by  a  period ;  this  will  serve  as  a  guide  in  forming  the  product 


§  VI.]  INTEGRATION  BY  PARTS.  79 

V  duy  which  is  to  be  written  under  the  integral  sign.     Thus,  let 
the  given  integral  be 


Lir^  log  X  dx. 


Taking  :!i^  dx  as  the  value  of  dv,  since  we  can  integrate  this 
expression  directly,  we  have 


,  dx 

x^  — 

3  3J       -^ 


x'^  log  X  dx  =  log  X'  —  x^ 

=  —x^  lo£r  X x^  dx 

3  "^  3J 

x^ 

=  -{3^ogx-  I). 

62.  The  form  of  the  new  integral  may  be  such  that  a 
second  application  of  the  formula  is  required  before  a  directly 
integrable  form  is  produced.  For  example,  let  the  given 
mtegral  be 

jt^  cos  X  dx. 

In  this  case  we  take  cos  x dx=  dv;  so  that  having  x^  =  u,  the 
new  integral  will  contain  a  lower  power  of  x:  thus 

x^  cos  X  dx  =  x^'s'm  X  —  2  Lr  sin  ;r  dx. 

Making  a  second  application  of  the  formula,  we  have 

Lit* cos xdx  =  x^s'mx—  2    x{-  cos x)  +    cos xdx 

=  jr'sin  X  +  2x  cos  x  —  2s\x\  x. 


8o  METHODS  OF  INTEGRATION.  [Art.  63. 

63.  The  method  of  integration  by  parts  is  sometimes 
employed  with  advantage,  even  when  the  new  integral  is  no 
simpler  than  the  given  one  ;  for,  in  the  process  of  successive 
applications  of  the  formula,  the  original  integral  may  be  repro- 
duced, as  in'  the  following  example: 

e''^^  sin  (nx  +a)dx 

=  ,,„. .  -  c°s  (^^  +  a)  ^  ^  U.^  ^^3  (^^  ^  „)  ^^ 
n  n]  ^  ' 

— V L  4.  _  e^^ V 1 ^„,x  sin  (fix  +  a)  dx, 

n  n  n  ft  ]  ' 

in  which  the  integral  in  the  second  member  is  identical  with 
the  given  integral ;  hence,  transposing  and  dividing, 

(^mx 
^mx  sii^  (ji^  ^  ^)  ^^  _  — g— — 2  \in  sin  {nx  +  a)  —  n  cos  (nx  +  a)]. 

64.  In  some  cases  it  is  necessary  to  employ  some  other 
mode  of  transformation,  in  connection  with  the  method  of 
parts.     For  example,  given  the  integral 

[sec^^^^; 

taking  dv  =  sqc^  6  dd,  we  have 

jsec8^^^  =  sec^.tan^- jsec^tan2  6'^6''.     .     .     (l) 


§VI.]  FORMULAS  OF  REDUCTION,  8 1 

If  now  we  apply  the  method  of  parts  to  the  new  integral,  by 
putting 

sec  d  tan  6  dS  =  dvy 

the  original  integral  will  indeed  be  reproduced  in  the  second 
member ;  but  it  will  disappear  from  the  equation,  the  result 
being  an  identity.  If,  however,  in  equation  (i),  we  transform 
the  final  integral  by  means  of  the  equation  tan^  6  =  sec^  0  —  i, 
we  have 

[sec^  ddd  =  sec  ^  tan  ^  -  jsec*  d  dO  +  [sec  Odd. 

Transposing, 

f      %  a  jn       sin  6       f   dd 

2   sec^  d  dd  =  — 2-^  4-    -; 

J  cos^  0      J  cos  ^ 

hence,  by  formula  {F),  Art.  31, 

fo  a  ,n         sin  ^        I  ,      ^      Vn      6^~| 
sec^  Odd  =  2-^  +  -  log  tan     -  +  -    . 
2  cos^  62^         L4      2  J 

65.  It  frequently  happens  that  the  new  integral  introduced 
by  applying  the  method  of  parts  differs  from  the  given  integral 
only  in  the  values  of  certain  constants.  If  these  constants  are 
expressed  algebraically,  the  formula  expressing  the  first  trans- 
formation is  adapted  to  the  successive  transformations  of  the 
new  integrals  introduced,  and  is  called  a  formula  of  reduction. 


82  METHODS  OF  INTEGRATION.  [Art.  65. 

For  example,  applying  the  method  of  parts  to  the  integral 


we  have 


x"" e^"^ dx  =  x"" \x''-'€^^dx,    .     .     .     .     (i) 


in  which  the  new  integral  is  of  the  same  form  as  the  given 
one,  the  exponent  of  x  being  decreased  by  unity.  Equation 
(i)  is  therefore  a  formula  of  reduction  for  this  function.  Sup- 
posing /^  to  be  a  positive  integer,  we  shall  finally  arrive  at  the 

8''-'^  dXf  whose  value  is — .     Thus,  by  successive  appli- 
cation of  equation  (i)  we  have 


I X"  £«^  dx  =  — 


n 

-  X"- 
a 


Reduction  of  Ism"'  6  dS  and  [cos'"  d  dO. 

66.  To  obtain  a  formula  of  reduction,  it  is  sometimes  neces- 
sary to  make  a  further  transformation  of  the  equation  obtained 
by  the  method  of  parts.     Thus,  for  the  integral 

[sin"'^^<9, 

the  method  of  parts  gives 

[  sin'«^^6'=  sin'«-^(9(-cos^)  +  (;;^  —  OJsin'^-^  ^cos^  <9^^. 


§  VI.]      REDUCTION  OF   TRIGONOMETRIC  INTEGRALS.  83 

Substituting  in  the  latter  integral  I  —  sin-  6  for  cos^  6, 
jsin'«  edd——  sin"'' -^  6^  cos  (9 

+  {fn  -  i)  I  sin"'- ==  6*  ^6*  -  {pi  -  i)  jsin'"  6'<^^; 
transposing  and  dividing,  we  have 

f  sin-  ede=^  ^'""'-  ^  '^"^  ^  +  ^^^^  fsin"-^  ede,  .   .   .   (d 

J  m  ml 

a  formula  of  reduction  in  which  the  exponent  of  sin  B  is  dimin- 
ished two  units.  By  successive  application  of  this  formula,  we 
have,  for  example : 

[    •     (,  n     in  sin'  B  cos  6  ^    [      .     »  n     m 

sm^ddd= g +  ^\  sm^Odd 

sin'/9cos  0      5  sin^/9cos  6      5  3  f  .  o  ^   ,/. 

— :: —   jr •     +    ^  -     Sm^  6  dd 

b  64  64J 

_       sin^  6*  cos  <9      5sin^<9cos<9      5.3sin(9cos^      5*3'^/? 
6  6-4  6-4-2  6-4-2 

67  By  a  process  similar  to  that  employed  in  deriving 
equation  (i),  or  simply  by  putting  6  =  ^7t  —  6'  m  that  equa* 
tion,  we  find 

f  „     ,^         cos"'-'  ^  sin  ^  m  —   I    [  ^„    ^  n  jn  /    \ 

cos'" 6 dd=. + cos''' -^6dd,    ,     .     (2) 

J  ;;/  ;;/      J 

a  formula  of  reduction,  when  in  is  positive. 


84  METHODS  OF  INTEGRATION,  [Art.  6%. 

68.  It  should  be  noticed  that,  when  m  is  negative,  equation 
(i)  Art.  66  is  not  a  formula  of  reduction,  because  the  exponent 
in  the  new  integral  is  in  that  case  numerically  greater  than  the 
exponent  in  the  given  integral.  But,  if  we  now  regard  the 
integral  in  the  second  member  as  the  given  one,  the  equation 
is  readily  converted  into  a  formula  of  reduction.  Thus,  put- 
ting —  n  for  the  negative  exponent  m  —  2,  whence 

m  =  —  n  +  2y 

transposing  and  dividing,  equation  (i)  becomes 

f  dO  cos^ 


n  —  2  f     dS  ,  . 


Jsin«(9  (/2—  i)sin''- 

Again,  putting  6  —  ^  n  —  6'  m  this  equation,  we  obtain 
{do  sm  6  n  —  2  {     dd 


Jcos"<9      (n  —  i)cos''-' 

Reduction  of  \sm"'e  cos""  0  dd, 

69.  If  we  put  dv  =  sin"'  6  cos  6  ddy  we  have 
cos«-^^sin"'+^/9 


n  —  2{dd  ,  . 

"^^"^^Jcos^^     •     •     •     •     (4) 


sin""  6  cos""  6  dd  = 


m  +  I 


+  -^ ^^- fsin'«+^6'cos«-^6>^^;  .     .     .     (l) 

m  +  I  } 

but,   if  in  the  same  integral  we  put  dv  =  cos«  6  sin  6  dd,  we 

have 

sin"'-^<9cos«+'  B 


j. 


sin""  6  cos""  e  dd 


n  +  I 


m L  fsin— 6'cos«+=^^^^.    ...    (2) 


§  VI.]       REDUCTION  OF   TRIGONOMETRIC  INTEGRALS.  85 

When  m  and  n  are  both  positive,  equation  (i)  is  not  a 
formula  of  reduction,  since  in  the  new  integral  the  exponent 
of  sin  6  is  increased,  while  that  of  cos  B  is  diminished.  We 
therefore  substitute  in  this  integral 

sin'«+^  6  =  sin'''  d{\  —  cos^  (9), 

so  that  the  last  term  of  the  equation  becomes 

^~  ^   fsin-  e  cos«-^  ddS-  ^LZJL  \  sin-  d  cos«  6>^6'. 
m  +  I  ]  m  +  I  ] 

Hence,  by  this  transformation,  the  original  integral  is  repro- 
duced, and  equation  (i)  becomes 


fi  + ^  1  f  sin-  d  COS"  ede:^ 


m  +  I 


n  —  I  ( 

+ sin-  dco^''-' Odd, 

m  -h  I  J 


Dividin    by  I  -^ = ,  we  have 

"  m  +  I       m  +  I 


sm-  6  cos''  6  dd=z  

m  -{■  n 


!^  fsin-6>cos''-^^^^,    ...     (3) 


+ 
m 


a  formula  of   reduction   by  which  the  exponent  of  cos  6  is 
diminished  two  units. 


86  METHODS  OF  INTEGRATION,  [Art.  69. 

By  a  similar  process,  from  equation  (2),  or  simply  by  put- 
ting Q  —  \n  —  d'  m  equation  (3),  and  interchanging  m  and  n^ 
we  obtain 

f   .       ^        ,  ^  ,^  sin'«-^l9cos«+^/9 

sm.*"  d  cos"  ddd  = 

J  '  m  +  n 

+  ^~  ^   fsin'«-^  d  cos«  ^^^,    .     .     .     (4^ 

a  formula  by  which  the  exponent  of  sin  6  is  diminished  two 
units. 

70.  When  7t  is  positive  and  m  negative,  equation  (i)  of 
the  preceding  article  is  itself  a  formula  of  reduction,  for  both 
exponents  are  in  that  case  numerically  diminished.  Putting 
—  mm  place  of  m,  the  equation  becomes 

fcos«  l9  _  cos«-^^  n  —  I    fcQs^-^  r/)  ,  >. 

J  sin"'  6      ~  ~  {m—  i)sm"'-^d      m  —  i  Jsin'«-^      •    •    •    •    i5; 

Similarly,  when  m  is  positive  and  n  negative,  equation  (2)  gives 

{'^de=^  '^r^    _^^£[giBn-%^.  ...  (6) 

Jcos«6'  (n- I)  cos''-' 6      n-ilcos^'-^d  ^^ 

7(.  When  m  and  ;2  are  both  negative,  putting  —  m  and  —  n 
in  place  of  m  and  ;^,  equation  (3)  Art.  69  becomes 


J  sii 


dd 


sin'''  ^  cos«  6  (m  +n)  sin'''- ^  (9  cos''+^  6 

n  +  I  f de^ 

m-^  n] sin'« d cos«+^ ^ ' 

in  which  the  exponent  of  cos  B  is  numerically  increased.     We 


§  VI.]      REDUCTION  OF   TRIGONOMETRIC  INTEGRALS.  8/ 

may  therefore  regard  the  integral  in  the  second  member  as  the 
integral  to  be  reduced.  Thus,  putting  n  in  place  of  »  +  2,  we 
derive 

f         dO I 

J  sin"'  6  cos«  d~  {n  —  i )  sin'«  -  ^  6  cos''  -  ^  6 

m  +  n  —  2  c  dd 


ZJ  f  ^^ (7) 

I      Jsin'«6'cos«-^^  '     ^^^ 


Putting  6  =  ^7t  —  d\  and  interchanging  ;/^  and  «,  we  have 

[__d^__  _ I 

J  sin'"  ^  COS"  ^  ~      (;«— i)sin'''~'6'cos''"'6^ 

m  +  n-2  f  dd 

m—\      Jsin'«-^(9cos"^ ^  ^ 


A 


72.  The  application  of  the  formulas  derived  in  the  preced- 
ing articles  to  definite  integrals  will  be  given  in  the  next  sec- 
tion. When  the  value  of  the  indefinite  integral  is  required,  it 
should  first  be  ascertained  whether  the  given  integral  belongs 
to  one  of  the  directly  integrable  cases  mentioned  in  Arts.  27 
and  28.  If  it  does  not,  the  formulas  of  reduction  must  be 
used,  and  if  m  and  n  are  integers,  we  shall  finally  arrive  at  a 
directly  integrable  form. 

As  an  illustration,  let  us  take  the  integral 

[sm^ecQ^^Bde. 

Employing  formula  (4)  Art.  69,  by  which  the  exponent  of  sin  % 
is  diminished,  we  have 


r  •  2  /I       X  fx  jn           sin  ^  cos"  Q      I  f       A 
sm^  Q  cos*  Odd  = ^ +  ^    cos* 


Odd. 


88  METHODS  OF  INTEGRATION,  [Art.  72. 

The  last  integral  can  be  reduced  by  means  of  formula  (2)  Art. 
6']^  which,  when  m  ~  /\^^  gives 

f  X  n  jn  COS^  ^  sin  ^         3    f         ^  n    in 

cos*  Qde=^  +  -    cos^  Bdd\ 

J  4  4J 

therefore 

r    .  9  /,       6.  n  jQ      sin  ^  cos^  6       cos^  d  sin  6      sin  6  cos  6  .    ^ 

I  sm^  e  cos*  6*  ^l9  = -. H + p +  —  . 

J  6  24  10  16 

73.  Again,  let  the  given  integral  be 

fcos^^ 
J     sin«6'    * 

By  equation  (5),  Art.  70,  we  have 

[cos^ddd  __  _    cos^/^  _  5  fcos*  d  do 
3    sin^  6    ~      2  sin^  6      2  J     sin  6'    ' 

We  cannot  apply  the  same  formula  to  the  new  integral,  since 
the  denominator  m—  i  vanishes  ;  but  putting  n—4.  and  m  —  —  i^ 
in  equation  (3)  Art.  69,  we  have 

cos^dde    cos^^     [cos^ede 


fcos*  ede  _  cos^^     r 

J    sin  6>     ~      2      "^  J 


3  J    sin  (9 


cos«^ 
3 


Jsin  e       J 


sin  ddO 


cos^  0  I 

+  log  tan  —6+  cos  6. 


3  2 

Hence 


[cos^  Odd  cos^  0       5  cos^  6*       5,  i  ^       S         /. 

I  — .  n  n  ■  = ^^-n  —  - — z log  tan  -  ^  —  ^  cos  0. 

J     sm^  6  2  sm^  (^  6  2^22 


§  VI.]  EXTENSION  OF   THE  FORMULA.  89 

Extension  of  the  Formula, 

74.  Let 

Y  W  dx  =z  ^,  (x), 

U^  {x)dx=  <f>,,(x)y 
etc.,     etc. ; 

then,  if  the  functions  ^^  (x),  (j),,  {x), ....  ^«  (x),  which  may  be 
called  the  successive  integrals  of  (l){x),  are  known,  and  also  the 
successive  derivatives  oi  f{x),  we  shall  have 

J/ W  -!>  W  dx  =  fix)  ^,  (.r)  -  j/'  (.r)  ^,(^)  ^^ 

=  fix)  <P,  {x)-J'{x)  I,  (x)  +  J/"  {x)  </,„  (x)  'dx. 

Continuing  this  process,  and  writing  for  shortness  /",  ^,,  . . .  for 

f{x)y  (j)^  {x)  .  .  .  we  have 

The  application  of  this  formula  is  equivalent  to  the  use  of  a 
formula  of  reduction.  Thus  the  value  of  x**  £^-^  given  in  Art.  65, 
may  be  derived  immediately  from  it. 


90 


METHODS  OF  INTEGRATION. 


[Art.  75. 


Taylor  s   Theore7n. 
75.  If,  in  the  formula  of  the  preceding  article,  we  put 
fi^x)  ^F'  {xo  +/i-  x),  and  (f>(^)  =  h 

Xo  and  h  being  constants, 
/'  {x)  =  -  F"  {xo+  h  -  x),        f"  {x)  =:  F'"  {Xo  +  h-  x),  etc.  ; 


and        (l>^{x)  =  x,       ^,X^) 


Hence 


x^  ;tr^ 

(j)    (x)  = ,  etc. 

1-2'  ^'"^    '  I-2-3 


[f'  {xo^h-  x) dx  =^F'  {xo+  h-  x)-x+  F"  (xo -^  h-  x)- 


x'^ 
2 


+ 


F''^'  (Xo  -^h-  x)     -- — dx. 


Now 


F'  {xo  +  /i  —  x)  dx  =  —  F  {xo  -{-  k  —  x); 


hence,  applying  the  limits  o  and  /i,  we  have 

F(x.  4-  /i)=  F{xo)+  F'  (Xo)  h  +  F"  (xo)  ^  + 

1  ^ 


Jo 


+  k  —  x) 


x"dx 
1-2.  •    n 


This  formula  is  Taylor's  Theorem,  with  the  remainder  expressed 
in  the  form  of  a  definite  integral. 


§VL] 


EXAMPLES. 


91 


/ 


/ 


/  y    I.      •svci-'^ X dxy 

•'  o 

y    2.     sec~^ji:^:r, 

J  o 

/    F  •    . 

i/  5.         Q  sin  Q^O, 

•'o 

V   6.        0  cos  mB  t/B, 

•'o 
J  o 

9.    Lcsec-^jpz/jf, 


Examples    VI. 


[  /V-^-V^ 


X  sin ~ ^  jc  +  V{i  —  X 


^  sec-^.v  —  log  [x  +  |/(.r'  —  i)]. 


;r  _  log  2 
4  2 


2»?  ^"^ 


-I  (  - 

tan-^x . 

2  2 


;r^f-^  —  2a:f-^  +  2£-^ 


i  [x"^  sec-^  A-  —  V{^^  —  i)]. 


V  10.   J'  S  sin    -  +  0   L/0,    —9  cos  ( -  +  6  j  +  sin  ( -+  6  j 


7r//2 


92 


METHODS  OF  INTEGRATION,  [Ex.  VI. 


/ 


.     X  sec"  X  dXy  X  tan  x  +  log  cos  x 

.    Lctan'x^/r     =  \x  (^qc^ x  —  i)  dx  L  xtsinx  +  log 


V     II 


12 


J   13.    Lv'sinx 


^JC, 


cos  X X  . 

2 


2X  Sm  X  -\-   2  cos  X  —  X  COS  jc. 


4.        x%m.~^xdxy  -x'*sm-'^x\ I    sin'' 6 ^0  =  —  . 

Jo  2  J,       2j^  8 

^  /  15.   Lr' tan- ^  ^  ^/jc, 


x'tan-^.r      x^      log  (i  +  ^) 
I  6    "^  ~6 


7t        2 


\l     16.        x'^mr^xdx,    -x'sin-^x-l ^(i  —  Jt:') 

.   Jo  o  9 

f^  (sin  X  —  cos  x)"!""  _  i 


17.  f-^COSJC^AT, 


T= 

2  2 

— lo 


\  J    18.    I  f-'^  *^"  ^  COS  jc  dx,  cos  /?  f^  t^"  ^  sin  (/?  +  x) 

19.     e-"^  sm^  xdx\  =  -    e-^  (i  —  cos  2x)  dx    , 


—  (cos  2x  —  2  sin  2x  —  k). 
10  ^  ^^ 


V  -I: 


f  ^  sin  6  </(9, 


e\  .  n14  .     I 

—  (sin  0  —  cos  6)       =  - 
2   ^  J.       2 


§  VI.]  EXAMPLES,  93 

£^  sin  X  cos  X  dXf  —  (sin  2j«:  —  2  cos  2x). 


sin'  fflQ  cos  ni^        36        3  sin  mh  cos  m^ 


22.    I  sin*/;z0^(9,  ^ +  ^ 

J^  '  4;^  8  Zm 

23.  Derive  a  formula  of  reduction  for  (log  x)^  x^  dx^  and  deduce 

J(log:r)^ 


from  it  the  value  of    {Xoo^xY  x^  dx. 


24. 


f                                                   x''"^^            n       f 
(log  xY  X'"  dx  =  (log  xy (log  x)»-^  x"^  dx. 

J  7/1'  ~i~   I  7/1  ~r~  I  J 

|(log^)'^V.r  =  (logx)'^  -  (log^)»  J'  +  HlM£  -  ?f.\    ! 

X  cos''  Jt' ^:v,  1^ X  sin :v  cos ^'  —  :i  sin*  x  +  ^x^. 

x^  sec  -  ^  :v      X  ^{x^  —  I )       log  [^  +  ^{x^  —  I )] 


/    25.       Ji:''  sec-  ^  X  dx^ 


26.  Derive  a  formula  of  reduction  for  La;'^  sin  (x  +  a)  dx^  and  de 
duce  from  it  the  value  of    x^  cos  x. 

\x^  sin  (^  +  «)  ^/;v  =  —  :v«  sin     x  -^  a  ■{ — 

+  n  \x^~'^  sin     a:  -f  o'  H —    dx, 
\x^  cos  X  dx  =^  [x^  —  20jr'  4-  120^)  sin  x  +  ($x*  —  6ar''  +  120)  cos:;*;. 


94 


METHODS  OF  INTEGRATION.  [Ex.  VL 


!7.     cos  6  sm  0  d^^    

ir 

f8.        COS*  0  sin*  6  ^/9, 


6  COS  0       sin  0  cos  0         I   r  •    .         ^1 

h  -7  [0  —  sin  0  cos  0  I. 

6  24  16 


32J0  512 

IT 

f4       4 .  7.  sin0cDs''0   ,   3sin0cos0  .   30 "14       8  +  3;r 

29.  cos*  0  //0, +  ^ ^ +  ^      = ^  . 

Jo  4  <>  ^Jo  32 

w 

30.  ^  COS*  0  ^/0, 


sin  0  COS  0  (8  COS*  0  +  10  cos' 0  +  15)  +  150 "Is  _  94/3  4-  i:>7r 
48  Jo  ~  96 


31 


32 


33 


f^os  0  . 
.     -^-5-  ^0, 
J  sin'  0     ' 

J  cos'  0     * 
fsin"  0  ^ 

IT 

(2COS'0    . 
^0, 
7rSin''0 

4 

^5-  j^(i  +  cos0r 

.6    \-^— 
^  '  Jsin0cos*0' 


cos'  0  _  3  cos  0  _  3  log  tan  ^Q 
2  sin^  02  2 


sm  0  sm  0         I 


4  cos  0        8  cos  G 


log  tan 


.4     2  J 


sm   0         5  sin"  0        K  ^  .  -, 

r—  — r  +  -  fS  —  sin  0  cos  0j 

3  cos   0       3  cos  02^  ■" 


cos-'0 
sin  0 


cos*  0 


48  -  15^ 
32 


1  [7    do'     ^ 

2  J„  cos*  0'  ~ 


I  1,0 

H r  +  log  tan - 


3  cos  0       cos 


§  VI.]  EXAMPLES,  95 

[  d^  I  3  cos  6      ,    3  ,       ^      Q 

'?7.    \— ^-s — ,  — ^^- — ■ c    '   2 h^  log  tan-. 

^'    J  sm  0  sm"  20 '  4  sm'  0  cos  6       8  sin'  0       8^2 

38.  Prove  that  when  n  is  odd 

H H +  log  tan  0  ; 


J  sin  0  cos''  0        /^  —  I  n  —  2> 

and  when  «  is  even 


f        //0  sec«-^0   ,   sec'^-'Q  ,  .  1  9 

-T— —^  = + H- 4-  log  tan  -  . 

J  sm  0  cos  0        n  —  I  n  —  $  2 

{       de  I  .   5  fsec"©    ,         ^  ,  ,       ^       e"1 

39-     -^-3 4-  ♦ ^-1 3-  + h  sec  0  +  log  tan  -     . 

^^    Jsin'0cos'o*       2Sin'0cos'0       2  [_     s  2  J 

4°-  Jy(.-,^_,)>     ^«^^^--sec0. 

^xVi^v'  -  i)  +  J  log  [:v:  +  V{x'  -  i)]. 

41.  J  (a'  -  xy  dx,  ^ '-^ -^  +  ^  sm-^  -  . 

a  5 

[  dx  \     U  .  ^TT  -^    Z 

42.  T^—, rTs  »  •  — T        COS    0  ^/0   =  ^^ a-  , 

(Jt:'  ^/a:  x^pc^  ~  ^)    I  tan~^  x 

'{x'  ^  i)"  8(x«  +  ir"^~~8   • 


96  METHODS  OF  INTEGRATION.  [Ex.  VI. 

,  f  cos's  —  sin'e    ,^  r       f/      ,     •    .        .\    cos 0  — sine     ,"1 

46.  \t-. \id^      =     (i  +  sin  0  cos  6)  7-r— -— ^d^    , 

^  J  (sin e  +  cose)'        L      J  ^(sme  +  cose)'     J' 


sin  e  cos  0 
sine  +  cose 


47.  Derive  a  formula  for  the  reduction  of  L%' sec**  a:  ^j<? ;  and  refer- 
ring to  Ex.  II,  thence  show  that  this  is  an  integrable  form  when  n  is 
an  even  integer.     Give  the  result  when  ;z  =  4. 


(j;sec«-2.rtanA:     sec^-^x 
X  sec"  X  dx  ■=^  — — — — —  —  , —^, r 
n  —  1                \n—  \)\n—  2) 

+  "-^^  i 

n  —  I  J 


X  sQC^-''x  dx. 


.       -    X  sec^  X  tan  x      sec'^  x      2  ^  1      ■^ 

X sec  X  dx  =  ' h  -  \^  tan x  +  log  cos  x\. 

3  63"- 

48.  Derive   a  formula  of   reduction  for    x  cos'*  x  dx,  and  deduce 
from  it  the  value  of    x  cos'  x  dx. 

J-         ,;i:cos'*-^:i:sin,r       cos**^      n  —  \  f 
X  cos«  ^  tf'.v  = 1 T. 1 \xQ.cys?*-^x  dx. 
n                       n              n      ] 


\         8     ^        .^sin.r  .    ,    cosJ»r 

Lr  cos  xdx  z:^ (cos''  X  ■\-  2)  ^ (cos^  x  +  6). 


49.   Find  the  area  between  the  curve 

y  =  sec  "  ^  .a;, 


§  VI.]  EXAMPLES.  97 

the  axis  of  x^  and  the  ordinate  corresponding  to  x  =  2. 

—  -  log  [2  +  ^3]  =  0.77744- 

o 

50.  Find  the  area  between  the  axis  of  x^  the  curve 
y  —  tan-  ^  x, 

7t         loff  2 

and  the  ordinate  corresponding  to  .^  =  i.         — ^—  =  0.43882. 


VII.     VjQ/^- 

Definite  Integrals, 

76.  Before  proceeding  to  transformations  of  definite  inte- 
grals involving  the  values  of  the  limits,  it  is  necessary  to 
resume  the  consideration  of  the  relations  between  a  definite 
integral  and  its  limits,  as  defined  in  the  first  section. 

By  definition,  the  symbol 


^  a 


dx 


denotes  the  quantity  generated  at  the  rate 

while  X  passes  from  the  initial  value  a  to  the  final  value  X. 
The  rate  of  x  is  arbitrary,  and  may  be  assumed  constant  ;  but 
in  that  case  its  sign  must  be  the  same  as  that  of  the  mcrement 


98  METHODS  OF  INTEGRATION.  [Art.  ^6. 

received  by  x ;  that  is,  the  sign  of  dx  is  the  same  as  that  of 
X-  a. 

These  considerations  often  serve  to  determine  the  sign  of 
an  integral.     Thus 

C  sin  xdx 


denotes  a  positive  quantity,  because  dx  is  positive,  and   — -^ 

is  positive  for  all  values  of  x  between  o  and  n, 

11.  Now  let  F  ix)  denote  a  value  of  the  indefinite  integral, 
so  that 

d{F{x)]^f{x)dx', 

thusy(;r)  is  the  derivative  of  F{x).  Then,  supposing  F  (x)  to 
vary  continuously  as  x  passes  from  a  to  ^;  that  is,  to  have  no 
infinite  or  imaginary  values  for  values  of  x  between  a  and  X, 
the  integral  is  the  actual  increment  received  by  F  {x)^  while  x 
passes  from  a  to  X.     In  this  case,  therefore 

f/(x)dx  =  F{X)-F{a) (I). 

J  a 

If,  on  the  other  hand,  there  is  any  value,  a^  between  a  and  X^ 
such  that 

F(pL)  =  ^^ 


equation  (i)  does  not  hold  true.     For  example, 

[dx  _ 


'dx  I 

X 


and  in  the  case  of  the  definite  integral 

f'    dx 


§  VII.]  DEFINITE  INTEGRALS.  99 

X  passes  through  the  value  zero,  for  which  F  {x)  is  infinite ;  we 
cannot  therefore  write 

C    dx _       I" 


=  —  2. 


This  result  indeed  is  obviously  false,  since  dx  is  here  positive, 
and  x^  Is  never  negative  for  real  values  of  x.  The  value  of  the 
integral  is  in  fact   infinite,  since  the    increments  received   by 

,  while  X  passes  from  —  i  to  o,  and  while  x  passes  from  o 

to  I,  are  both  infinite  and  positive. 

78.  Since  the  derivative  of  a  function  becomes  infinite  when 
the  function  becomes  infinite JDiff.  Calc,  Art.  104;  Abridged 
Ed.,  Art.  89],  v/e  can  have  F  (n)  =  00  only  when  /"(^)  =  00  ; 
but  it  is  to  be  noticed  that  F(x)  does  not  necessarily  become 
infinite  when/(;i')  becomes  infinite.     Thus,  in 

r^  dx 

f{x)  —  x~\^  which  becomes  infinite  for  ;r  =  o,  a  value  of  x 
between  the  limits ;  but  since 

iv-^dx^ix^ 

the  indefinite  integral  F(x)  does  not  become  infinite.  There- 
fore equation  (i)  holds  true,  and 


J-X^3  2  J_, 


79.  We  have,  in  the  preceding  articles,  assumed  that  the 
independent  variable  varies  uniformly  in  passing  from  the 
lower  to  the  upper  limit ;  but  when  a  change  of  independent 
variable  is  made,  the  new  variable  does   not   generally   vary 


100  METHODS  OF  INTEGRATION.  [Art.  79. 

uniformly  between  its  limits.  It  is,  however,  obvious,  that,  in 
equation  (i).  Art.  'JJ,  x  may  vary  in  any  manner  whatever  in 
passing  from  a  to  X^  provided  that  F{x)  remains  throughout 
a  continuous  one-valued  function ;  x  may  even  pass  through 
infinity,  provided  F  (x)  is  finite  and  one-valued  when  ;ir  =  00  . 


Multiple-  Valued  Integrals, 

80.  When  the  indefinite  integral  is  a  multiple-valued  func- 
tion, a  particular  value  of  this  function  must  of  course  be 
employed,  and  it  is  necessary  to  take  care  that  this  value  varies 
continuously  while  x  passes  from  the  lower  to  the  upper  limit. 
In  the  fundamental  formula  (7)  it  is  sufficient  (provided  the 
radical  V(i  —  x^')  does  not  change  sign),  to  limit  the  meaning  of 
the  symbols  sin-';i:  and  cos"^;r  to  the  primary  values  of  these 
symbols  (see  Diff.  Calc,  Arts.  54  and  55),  since  these  values 
are  so  taken  as  to  vary  continuously  while  x  passes  through 
all  its  possible  values  from  —  i  to  +  i. 

81.  In  the  case  of  formula  (k)  the  primary  value  of  tan-'  x 
is  so  defined  that,  as  x  passes  from  —  00  to  +  00 ,  the  primary 
value  varies  continuously  from  —  ^-tt  to  +  \n.  We  may  there- 
fore employ  the  primary  value  at  both  limits,  unless  x  passes 
through  infinity,  as  in  the  following  example.  Given  the  inte- 
gral 

ff  de  ff  SQC^edd 


[6    se< 
Jo  '1  + 


J„  cos^i^  +  gsm^e      Jo  I  +  gtdiVL^e' 
if  we  put  tan  6  =  Xy  this  becomes 


3       dx  I 


tan-^3;i;       ^  =  -  [tan-^(— V3)  —  tan-^  o], 
— lo  3 


Jo      1+9^       3 

But  here  it  is  to  be  noticed,  that,  as  6  passes  from  o  to  |-7r,  x 


§  VII.]  MULTIPLE-VALUED  INTEGRALS.  101 

passes  through  infinity  when  6  =  \7i.  Hence,  if  the  value  of 
tan-'3;i'  is  taken  as  o  at  the  lower  limit,  it  is  to  be  regarded  as 
increasing  and  passing  through  \n^  when  ;ir  =  oo ,  so  that  its 
value  at  the  upper  limit  is  |;r,  and  not  —  \n.     Hence 

f6  dd  27t 


cos^  6^  +  9  sin^  8 


82.  When  the  symbol  cot-^  x  is  employed,  the  primary 
value,  defined  in  the  same  manner  as  in 'the  case  of  tan"':r, 
cannot  be  taken  at  both  limits  when  x  passes  through  zero. 
Thus,  using  the  second  fo'rm  of  (>^),  Art.  10,  we  have 

=  cot"'  I  —  cot"^  (—  i), 


Jx    I  +^ 

in  which,  if  cot"'  i  is  taken  as  J  ;r,  cot-'(—  i)  must  be  taken 
as  }  n.     Thus 

f"'    dx  I 


I    I  +  ;r^  2 


Formulas  of  Reduction  for  Definite  Integrals, 

83.  The  limits  of  a  definite  integral  are  very  often  such  as 
to  simplify  materially  the  formula  of  reduction  appropriate  to 
it.     For  example,  to  reduce 


j: 


x""  ^-""dx, 
we  have  by  the  method  of  parts 


102  METHODS  OF  INTEGRATION,  [Art.  83. 

Now,  supposing  n  positive,  the  quantity  ;tr'' £--^  vanishes  when 
x  =  Oy  and  also  when  ;r  =  00  [See  Diff.  Calc,  Art.  107 ;  Abridged 
Ed.,  Art.  91].     Hence,  applying  the  limits  o  and  00 , 


x^'B-'' dx  =  n  I   x""-"-  e-^'dx. 

By  successive  application  of  this  formula  we  have,  when  n  is 
an  integer, 

^n  e—r ^^  _  ^  ^^  _  jj 2.1. 

Jo 

84.  From  equation  (i)  Art.  66^  supposing  m  >  i,  we  have 

■n  IT 

[  sm^^  d  dd  =  ^^^—^  [  sin--  ddO. 

Jo  ^         Jo 

If  m  is  an  integer,  we  shall,  by  successive  application  of  this 

n  IT 

formula,  finally  arrive  at  V  dd  =  -  or  j'  sin  ^^^  =  i,  according 
as  m  is  even  or  odd.     Hence 

if  m  is  even,      f  sin"  6  dO  =  (^j)(>«  -  3)  •  ■  •  •  i  .  5,  .  .  (p) 
Jq  m{m  —  2) 2    2  ^     ^ 

and  if  m  is  odd,       f^  sin-  d  dO  =  {m  -  i){m  -  3)  >  -  -  •  2    _  (p^ 
Jo  m{7n-2) I         /     ^ 


§  VII.]  FORMULAS  OF  REDUCTION.  IO3 

85.  From  equations  (3)  and  (4)  Art.  69,  we  derive 

l  sin"^  e  cos«  ede  =   ^  ""  ^  f'  sin'''  6  cos"-^edd, 

Jo  m  +  n  io  -  _ 

IT  JT 

and  f'  sin'^  6  cos'^  ^  ^<9  =  ^-^LzJL  y  sm*"-^  d  cos«  0  dd, 

Jo  ^^  +  n  Jo 


By  successive  application  of  these  formulas,  we  shall  have  for 
the  final  integral  one  of  the  four  forms 


\^  dd,         Kin^^^,       Kos^^^,  or  [' sin  ^  cos  ^^^. 

Jo  Jo  Jo  "O 

The  numerator  of  the  final  fraction  ( or )  is  in  each 

case  either  2  or  i.  In  the  first  case,  the  value  of  the  final  inte- 
gral is  J  7t,  and  the  final  denominator  is  2 :  in  the  second  and 
third  cases,  the  value  of  the  final  integral  is  i,  and  the  final 
denominator  is  3 :  in  the  fourth  case,  the  value  of  the  final 
integral  is  J,  and  the  final  d.enominator  is  4.  Therefore  (since 
the  factors  in  the  denominator  proceed  by  intervals  of  2),  it  is 
readily  seen  that  we  may  write 

F sin'«  6  cos«  6 dO  =  (^^-i)(^- 3)  •->  (^- 0(^:^3)^ ,,  .  (g) 


provided  that  each  series  of  factors  is  carried  to  2  or  i,  and  a  is 
taken  equal  to  unity,  except  when  m  and  n  are  both  even,  in  which 
case  a  =  ^  Tt. 


104  METHODS  OF  INTEGRATION.  [Art.  86. 


Elementary  Theorems  Relating  to  Definite  Integrals, 

86.  The  following  propositions  are  obvious  consequences  of 
equation  (i^,  Art.  T^j. 


^f{x)dx=-^f{x)dx (I) 

(f{x)dx=\f{x)dx+(f{x)dx.     .     .     (2) 

i  a  J  a  J  c 

Again,  if  we  put  x  ^  a  ■\-  b  —  z^wq  have 

I"  f{x)dx  =  -  {  f(a  -V  b-  z)  ds  =[  f{a  +  b-z)dz 


by  (i),  or  since  it  is  indifferent  whether  we  write  ^  or  ;i;  for  the 
variable  in  a  definite  integral, 


\f{x)dx=    \/(a-hb-x)dx     ....     (3) 

l{  a=c,  we  have  the  particular  case 

^'/{x)dx=^'/{b-x)dx     ....     (4) 


§  VII.]  DEFINITE  INTEGRALS.  IO5 

87.  As  an  application  of  formula  (4),  we  have 

■K  It  1t_ 

V  COS-  ede=^  cos'«(^  -e\de=  ["  sin-  ddO     .     .     .     .     (i) 

IT  V 

Hence  the  value  of    ^  cos'"  8  dO  as  well  as  that  of    ^  sin"'  6  dS 


is  given  by  formulas  {P)  and  {P').  The  values  of  these  integrals 
are  readily  found  when  the  limits  are  any  multiples  of  ^  n. 
For,  by  equation  (2)  of  the  preceding  article,  we  may  sum  the 

values  in  the  several  quadrants.     But,  putting  6  =^  k — h  ^',  and 

employing  equation  (i),  we  have 

'sm-^ede=±\        '  cos-'dd0=±\    sin-^dde,   .   . 


(2) 


in  which  the  sign  to  be  used  is  determined  by  that  of  sin'«  6 
or  cos'"  6  in  the  given  quadrant. 

In  like  manner  the  value  of  the  integral  in  formula  (Q)  is 
numerically  the  same  in  every  quadrant,  and  its  sign  is  the 
same  as  that  of  sin'"  ^cos'^  ^in  the  given  quadrant. 


Change  of  Independent  Variable  in  a  Definite  Integral, 

88.  It  is  often  useful  to  make  such  a  change  of  independ- 
ent variable  as  will  leave  unchanged,  or  simply  interchange, 
the  values  of  the  limits.  As  an  illustration,  let  us  take  the 
definite  integral 


f- 

Jo  I   + 


I06                              METHODS  OF  INTEGRATION.                  [Art.  88 
If  we  put  X  —  — ,  whence  log  x  =  —  log  j,  and  dx  — ^» 

y  r 


•o 

u  = 

.    00 

logj 

f+y  +  l 

dy  = 

-u; 

whence 

we  infer  that 

•  00 

u  = 

,  o 

log;r 

dx  = 

0. 

89. 

Again,  let 

•  00 

u  = 

^^'- 

Putting  ;ir  =  — ,  we  have 

_  r2iog^- 

'°g-^^r- 

1   !(->£ 

..r. 

dy 

Jo         a^^f  •"  ^      )oa^-\-f 

hence 

flog 


log     X  y  _7t    lOg^ 


;r*  2a 


Differentiation  of  an  Integral, 


90.  The  integral 


f  {x)  dx  is  by  definition  a  function  of  x. 


whose  derivative,  with  reference  to  jr,  is  f{x).     Thus,  putting 

U=  \f{x)dx, 

i  a 

dU      ,,  , 


g  VII.]  DIFFERENTIATION  OF  AN  INTEGRAL.  lO/ 

This  gives  the  derivative  of  an  integral  with  reference  to  its 
upper  limit.     By  reversing  the  limits  we  have,  in  like  manner, 

when  the  lower  limit  is  regarded  as  variable, 
91.  Now  writing  the  integral  in  the  form 


U 


\    u  dx  ^ (i) 


if  u  is  a  function  of  some  other  quantity,  «',  independent  of  x 
and  Uy  U  \s  also  a  function  of  a^  and  therefore  admits  of  a  de- 
rivative with  reference  to  a.     From  (i)  we  have 

dJJ__ 
dx~^' 
whence 

d  dU  _  dn, 
da  dx        da 

By  the  principle  of  differentiation  with  respect  to  independent 
variables  [See  Diff.  Calc,  Art.  401 ;  Abridged  Ed.,  Art.  200]. 

d^dU^d_  dU 
dx  da       da  dx  ' 


Therefore 


and  by  integration 


d  dU  _  du  ^ 
dx  da       da ' 


dU      {du   J         ^  /  X 

dx  +  C (2) 


dU  __  (du 
da  ~  ]da 


108  METHODS  OF  INTEGRATION.  [Art.  9I. 

Now,  in  equation  (i),  C/ is  a  function  of  x  and  a  which,  when 
;r  =  «,  is  equal  to  zero,  independently  of  the  value  of  a.  In 
other  words,  it  is  a  constant  with  reference  to  a^  when  x  —  a\ 

therefore  -r-  —  o  when  x  =^  a.     If,  then,  we  use  ^  as  a  lower 
da 

limit  in  equation  (2),  we  shall  have  (7  =  0.     Therefore 


dU 

da 


du  J  ,  . 


Substituting  for  x  any  value  b  independent  of  a^  we  have 

---\    udx —  \    -j-u dx , (4) 

aai  a  ]  a  da 

which  expresses  that  an  integral  may  be  differentiated  with 
reference  to  a  quantity  of  which  the  limits  are  independent^  by 
differentiating  the  expression  under  the  integral  sign. 

92.  By  means  of  this  theorem,  we  may  derive  from  an  inte- 
gral whose  value  is  known,  the  values  of  certain  other  inte- 
grals.    Thus,  from  the  first  fundamental  integral, 


x»dx  =  - , (l) 


we  derive,  by  differentiating  with  reference  to  n, 


_  (;2  +  i)x''-^^\ogx  —x""-^^ 
{n  +  1/ 


X"  log  X  dx  =  /  ..    ,     ,  \2 


the  result  being  the  same  as  that  which  is  obtained  by  the 
method  of  parts. 

93.  The  principal  application  of  this  method,  however,  is 
to  definite  integrals,  when  the  limits  are  such  as  materially  to 


§  VII.]  DIFFERENTIATION  OF  AN  INTEGRAL.  IO9 

simplify  the  value  of  the  original  integral.     Thus,  equation  (i) 
of  the  preceding  article  gives 


X''  dx  —  — ^—  , 
;2+  I 


1: 

whence,  by  successive  differentiation, 


I    x^  loff  X  dx=  —  7 ^ , 

Jo  («  +  0' 


1-2 


'^dx^  -, ^, 


I  x^'ilogx) 

[  x"(\ogxYdx=  (-  lY^'^"  "'' 


Integration  under  the  Integral  Sign, 

94.  Let  u  be  a  function  of  jt  and  a,  and  let  a  and  ao  be  con- 
stants ;  then  the  integral 

U=\     r[   udA^da, (i) 

is  a  function  of  x  and  ^,  which  vanishes  when  a  — 0.0^  inde- 
pendently of  the  value  of  x^  and  when  x  =  a.,  independently  of 
the  value  of  a.     From  (i) 


dU      [        ,  ^  d  dU 

whence  -y-  3— 

a  ax  eta 


dU      [       , 
—  -=1  \    u  ax, 
da       Ja 

therefore      -—-—  =  «,  whence  -=-  =  \u da  -^  C, 

dadx  dx        J 


no 


METHODS  OF  INTEGRATION-. 


[Art.  94. 


Now  -%—  must  vanish  when  a  =  a^,  since  this  supposition  makes 

^independent  of  x;  therefore,  if  we  use  ^^^  for  a  lower  limit 
in  the  last  equation,  we  must  have  C  =  0;  therefore 


dx 


—       u  da^ 


and  since  u  vanishes  when  x  =  a, 

U  —  \        \    u  da 

ia  VJ  OL^ 

Comparing  the  values  of  6^  in  equations  (i)  and  (2),  we  have 


dx. 


(2) 


tc  dx  da  = 

aja  }a}a 

It  is  evident  that  we  may  also  write 

2L  dx  da  — 
ot-Jci  ]a]a 


u  da  dx. 


■  (3) 


provided  that  the  limits  of  each  integration  are  independent 
of  the  other  variable. 

96.  By  means  of  this  formula,  a  new  integral  may  be  de- 
rived from  the  value  of  a  given  integral,  provided  we  can  inte- 
grate, with  reference  to  the  other  variable,  both  the  expres- 
sions under  the  integral  sign  and  also  the  value  of  the  inte- 
gral.    Thus,  from 


x""  dx  = 


n  +  I 


§  VII.]      INTEGRATION   UNDER    THE  INTEGRAL    SIGN.         Ill 


by  multiplying  by  dn,   and  integrating  between  the  limits   / 
and  5,  we  derive 

whence 

—^ ax  —  lop^ . 

J^       log  A'  ^7-+    I 

96.  When  the  derivative  of  a  proposed  integral  with  refer- 
ence to  rt'  is  a  known  integral,  we  can  sometimes  derive  its 
value  by  integrating  the  latter  with  reference  to  vc.     Thus,  let 


u  — 


' — dx (I) 


In  this  case 


da       io  "  «'    Jo  oc^ 


hence,  integrating,        u=—  log  ^^  +  6^  =  log  —     .     .     .     .  (2) 
since  in  (i)  u  vanishes  when  a—fi. 


The  Definite  Integral  Regm^ded  as  the  Limiting  Value 

of  a  Sttm. 

97.  Let  A  denote  the  greatest,  and  B  the  least  value  as- 
sumed by/(jt'),  while  x  varies  from  a  to  b.  Then  it  is  evident 
that 

t  f{x)dx<  f  Adx; (i) 

J  a  J  a 

for,  while  x  passes  from  a  to  b,  the  rate  of  the  former  integral 


112  METHODS  OF  INTEGRATION.  [Art.  97. 

is  generally  less,  and  never  greater  than  the  rate  of  the  latter. 
In  like  manner 

fb                    fb 
f{x)dx>  \    Bdx (2) 

J  a  J  a 


The  values  of  the  integrals  in  the  second  members  of  equations 
(i)  and  (2)  are  A  {b  —  a)  and  B  {b  —  a)  respectively.  There- 
fore, if  we  assume 

(f{x)dx  =  M(b-d), (3) 

we  shall  have  A  >  M>  B. 

The  quantity  M  in  equation  (3)  is  called  the  mean  value  of  the 
function /(jr)  for  the  interval  between  a  and  b. 
98.  Let 

b—  a  =  n  Ax\ (4) 

then  the  n  +  i  values  of  x, 

a,  a  +  AXy  a-i-2Ax,--''         b, 

define  n  equal  intervals  into  which  the  whole  interval  b  —  a  is 
separated.  Let  x^^  x^, x„hG  n  values  of  x^  one  com- 
prised in  each  of  these  intervals;  also  let  2^/{Xr)  Ax  denote 
the  sum  of  the  n  terms  formed  by  giving  to  r  the  n  values 
I  •  2  •  •  •  •         n  in  the  typical  term/(;ir^)  Ax;  that  is,  let 

2i/{xr)  Ax=/{x,yAx -i-/{x^)    AX""-{-/{x„)  AX,   .     .    (5) 


§  VII.]  AN  INTEGRAL    THE  LIMIT  OF  A    SUM.  II3 

We  shall  now  show  that  when  n  is  indefinitely  increased  the 
limiting  value  of  ^f/(^r)  A;r  is     fix)  dx. 

99.  If  we  separate  the  integral  into  parts  corresponding  to 
the  terms  above  mentioned  ;  thus, 

Jb  ta  +  AJtr  /*«  +  2  A  J* 

/{x)  dx  =  f{x) dx  +  f{x)dx  • .  . . 

+  f  /{x)dx, 

and  let  J/j,  M^,  •  •  •  •  Mn  denote  the  mean  values  of  f  (x) 

in  the  several  intervals,  we  have,  in  accordance  with  equation 
(3),  Art.  97, 

I  f{x)  dx  =  Af^Ax  -^M^  AX -h  M„  AX (6) 

J  a 


Now,  since  /(-tv)  and  Mr  are  both  intermediate  in  value 
between  the  greatest  and  the  least  values  of  /{x)  in  the  inter- 
val to  which  they  belong,  their  difference  is  less  than  the  dif- 
ference between  these  values  of /(;t').     Therefore,  if  we  put 

/{Xr)  =  Mr   +   er, (7) 

er  is  a  quantity  whose  limit  is  zero  when  n,  the  number  of 
intervals,  is  indefinitely  increased,  and  A;r  in  consequence 
diminished  indefinitely. 

Comparing  the  terms  in  equations  (5)  and  (6)  we  have,  by 
means  of  equation  (7), 

2l/{x)  AX  =     f{x)  dx  +  (e^-{-  e^ +  en)  t.x.      ...      (8) 


114  METHODS   OF  INTEGRATION.  [Art.  99. 

Denote  by  e  the  arithmetical  mean  of   the  n  quantities  ^1, 
^2»  •  •  •  •  ^« ;  that  is,  let 


«£  =  ^1  +  ^2  +  ^3 ^« ; (9) 


then,  since  e  is  an  intermediate  value  between  the  greatest  and 
the  least  value  of  ^^,  it  is  also  a  quantity  whose  limit  is  zero 
when  n  is  indefinitely  increased.  By  equations  (9)  and  (4), 
equation  (8)  becomes 


b  [^ 

2^  f{x,)  t^x  =  Y^  f{x)  dx  +  e{b-  a\ 


whence  it   follows   that     f(x)  dx  is  the  limit  of  ^^/  (^v)  dx 

J  a 

w^hen  n  is  indefinitely  increased,  since  the  limit  of  c  is  zero. 

100.  It  was  shown  in  the  Differential  Calculus,  Art.  390 
[Abridged  Ed.,  Art.  193],  that,  in  an  expression  for  the  ratio 
of  finite  differences,  we  may  pass  to  the  limit  which  the  ex- 
pression approaches,  when  the  differences  are  diminished  with- 
out limit,  by  substituting  the  symbol  d  for  the  symbol  A. 
The  theorem  proved  in  the  preceding  articles  shows  that,  in 
like  manner,  in  the  summation  of  an  expression  involving 
finite  differences,  we  may  pass  to  the  limit  approached  when 
the  differences  are    indefinitely   diminished,  by  changing   the 

symbols  ^  and  A  into     and  d. 

The  term  integral,  and  the  use  of  the  long  s,  the  initial  of 
the  word  sum,  as  the  sign  of  integration,  have  their  origin  in 
this  connection  between  the  processes  of  integration  and  sum- 
mation. 


VII.]     ADDITIONAL  FORMULAS  OF  INTEGRATION.  II5 


Additional  Formulas  of  Integration. 

I0(.  The  formulas  recapitulated  below  are  useful  in  evalu- 
ating other  integrals.  {A)  and  {A')  are  demonstrated  in 
Art.  17;  {B)  and  {C)  in  Art.  29;  {D)  and  {E)  in  Art.  30; 
{F)  in  Art.  31  ;  (6^)  and  {G')  in  Art.  35  ;  (//)  and  (/)  in  Art.  50 ; 
{7)  in  Art.  51  ;  {K)  in  Art.  52  ;  {L)  in  Art.  53  ;  {M)  in  Art.  55  ; 
(N)  and  ((9)  in  Art.  58;  {P)  and  (P')  in  Art.  84;  and  (0  in 
Art.  85. 


b)      a-b 


log 


X  —  a 


dx 


f     <ar.r  i  ,      x  —  a 


,^ 


2a 


X  -[-  a 


.      (A) 
.   {A') 
.     .   {B) 


{sm^ectO  =  i{d  -sin  OcosO).     ........ 

[cos2^^(9  =  -|(^+ sin6>cos^).     .,..,.....  ((fl 


J  sin  6^  c( 


cos  u 


logtan^.     .    ^ ,,:.(/;) 


C  dd        ,       ^       .  ^       ,       I  —  cos  ^ 
- — 7,  =  Iog:  tan  lu  —  log r — 7^ — 


^  ==  log  tan     -  +  - 

J  cos  (9  ^  L4      2 

L 


+  bcosd       Via^~^) 


r  tan 


log 


I  +  sin  ^ 
cos  6 

a-b 


J   a  ^  b 


tan 


,.]. . . 


Il6  METHODS   OF  INTEGRATION.  [Art.   lOI. 

[        dB         __  I  \/{b^a)  ^  V{b  -d)t^n\d 


— -— ^  —  -\og-^ '- .    .......    {H] 


-        log  ^~ \       o (/) 


X  V{d^  —  x^)       a 
dx 


'.     (?) 


\^s/(x^  ±  d')dx=^^^'^^  ""'^   ±  l'  log  [x  +  V{^±a')-\ .    .    (L) 
'  (a^  —  x^)  dx  =  —  sm-^-  -\ — {M) 


dx 


dx 


2sin"^ 


/^^ (O) 


Jo  Jo  mint  —  2) 2    2  ^    ^ 


§  VII.]      ADDITIONAL  FORMULAS  OF  INTEGRATION.  II7 

Jo  Jo  m(ni-2) I  ^     ^ 

(m  +  n){m  +  n  —  2) ^"^^ 


in  which  a=  i,  unless  m  and  n  are  both  even,  when  a  =  ~, 

2 


Examples  VII. 

I-        — T"! ;>         [^  >  ^>  ^^^  ^  ^"  integer]  -— — ^^ r^. 

p"^^±-         do  2n7t±\7t 

^*    Jo  2  +  COS  9'  V3 

IT 

3.  r  sin"  ear©, 

4.  sin*  0  fl?i9, 

5.  J    „cos'0^e, 


5^ 
32" 

16 
15' 


6.    I  sin^  0  cos®  6  //G,  512 


Il8  METHODS  OF  INTEGRATION,  [Ex.  VIL 

7.    I  sin'  0  cos'  ^d%  ^ 


'    Jo  4/(1-^')' 
'"  J. 


35 


sin'«  Q  dQ , 


8.     ^  sin"^  e  cos"'  G  ^9,  —  I   si 

Jo  2'«Jo 


f      x'^"  dx ^  1*3*5   •   •   ■   (2^  —  1)  Ti" 

Jo  4/(1  — ^"7*  2-4-6  •   •   •   •   2«     2 


2'4'6-   ■    '   '  2n 

4/(1--^')'  3.5.7.     .     .    (2«+l)* 


2tf^ 


^^' '  63- 


f    {x'~a')'^ax  ,  3;r 

1  ^  •  I  R  »  7~ 

I0« 


r    x' dx  8 

'^-  JoU^  +  ;.')r 


15.  Prove  that 

o  o 

and  derive  a  formula  of  reduction  for  this  integral,  supposing  «  >  0 
and  m  '>  1. 

o  n     Jo 


§  VII.]  EXAMPLES,  119 

16.  Deduce  from  the  result  of  Ex.   15   the  value  of  the  integral 
when  m  is  an  integer. 


Jo  n\n-\-i)   '   '   '[n  +  m—i) 

17.  Wa+xY(a-xy  ^x.     See£x.i6.  2^V2>^ 

J  -a 

It 

18.  Tsin'  Q  (cos  6)^  do.     Fut  sin'^  G  —  x,  and  see  Ex.  16. 


4504s 


5-7. II. 19 
19.  Show  by  a  change  of  independent  variable  that 

r      x^  dx       _  r       a'  dx 

Jo    (a"  +x^Y  -Jo  {a'  +xy   ' 

^  ^    ,  r   x'  dx       I  r   dx       n 

and  therefore  -7-^— — ^vi  —  ~      t—. — i  =  —  • 

Jo  («  +  x^y       2  jo  a    +  x""      4a 

(""xjogx^dx  log  dJ 

Jo     V^   +^  )  2df 

r  tan-^v.  dx  7t^ 

^^'    ]^x'  ^x-\-  1'  '  '  6^3' 

r         .  X       xdx  7t* 

22.  tan"^ —     4,4,  -^-7- 
Jo            a  x^  -\-  a*'  i6a' 

23.  Derive  a  series  of  integrals  by  successive  differentiation  of  the 
definite  integral  |    f"^  dx. 

r  .         1-2'  '-n 

X^    E-*^  dx  =  ; . 

Jo  ««-^   ' 


120  METHODS  OF  INTEGRATION,  [Ex.  VII. 


m 


24.  Derive  from  the  result  of  Art.  d"^  ^^  definite  integrals 

(°°                                      „  f  > 

B  -  '"■*  sin  nx  dx  =  —. 5 ,     and  f-  '«^  cos  nx  dx  = 

o  m   -{■  n  '  Jo 

and  thence  deri .  e  by  differentiation  the  integrals 

^xe-  "-sin  nxdx  ^  -^-.—-^^^  and  J^  xe-  --cos  nx  dx  =  ^^._^^.^ 

25.  From  the  results  of  Ex.  24  derive 


i:- 


.«,,-•  ^         2n{sm  —  n) 


2       1        „2\l»         » 


{m'  +  7i') 


1: 


„  ,         2m  (m  —  3n) 

x^  £-  '"^  cos  nx  dx  =  — 7-H ^Ts"  • 

(m'  +  n) 

26.  From  the  fundamental  formula  (k')  derive 


(dx  _         TT 


and  thence  derive  a  series  of  formulas  by  differentiation  with  refer- 
ence to  a. 


dx  71       1-3  ••  •  (2«  —  3)       I 


27.  Derive  a  series  of  integrals  by  differentiating  with  reference  to 
/5,  the  integral  used  in  Ex.  26. 

p    x^'^-^dx    _     7t    i'3-5  >  •  (2;^  —  3)        I 
Jo  (a  +  l^x'V  ~  2«a:i  i-2-3~-  •  •  («  -  i)  'yS«-*  * 


§  VIIJ  EXAMPLES.  121 

28.  From  the  integral   employed   in  examples  26    and  27,   derive 

the  value  of  -. ; — Tr-arr  • 

Jo  (n;  +  ^^  )  ^ 

Differentiate  tivice  with  reference  to  /?,  ««^  ^«^^  a///^  reference  to  a. 

fx* dx         __  i-3'i  re 

29.  Derive  an  integral  by  differentiation,  from  the  result  of  Ex,  II.,  67 

Jo  (^-^  +  ^0  (^'  +  ay  ~  ^a'b  {a  +  bf  ' 

30.  Derive  an  integral  by  integrating  —, ^  =  — . 

J  o  a    ~T'  X        2a 

fTtan-.^-tan-.^l^  =  ^log^. 

JoL  ^  xj  X        2      ^  g 

31.  Derive  a  definite  integral  by  integrating 


1:- 


sin  nx  ax  =  —5 5 

/«    +  n 

with  reference  to  n. 


m^  +^' 


(cos  «:\:  —  cos  bx)  ax  —  —  \oa^ 

]o    X  2        m 

32.  Derive  a  definite  integral  from  the  integral  employed  in  Ex.  3: 
by  integration  with  reference  to  m. 


22  METHODS  OF  INTEGRATION.  [Ex.  VIL 

T^2,'  Derive  an  integral  by  integrating  with  respect  to  m 


ffi 
€-  ^"^^  COS  nx  dx  =  —T, — 


^ COS  nx  ax^=  —  log  -^ 


34.  Derive  an  integral  by  integrating  with  respect  to  n  the  integral 
used  in  the  preceding  example. 


re-'"^  .  .  .     .   X  ^  m(a-  b) 

■  (sin  ax  —  sin  bx)  dx  =  tan" '  — ^ / 

Jo     ^^  ^  m'  +  ab 


m^  +  ab  ' 
35.  Show  by  means  of  the  result  of  Ex.  32  that 


(•00  • 


sm  nx  .  TT 
ax  =  — 

X  2 


$6.  Derive  an  integral  by  integration  from  the  result  of  Ex.  II.,  67. 

(CO                   2       1         '  2 
log    2 ^dx  by  the  method  of  Art.  96. 

7t(a  —  b). 
38.  Evaluate       log     i  +  — ,    logxdx.  it  a  (log^  ~  i). 


§V1II.]  PLANE  AREAS.  I23 

CHAPTER   III. 

Geometrical  Applications. 

VIII. 

Plane   Areas, 

102.  The  first  step  in  making  an  application  of  the  Inte- 
gral Calculus  is  to  express  the  required  magnitude  in  the  form 
of  an  integral.  In  the  geometrical  applications,  the  magni- 
tude is  regarded  as  generated  while  some  selected  independ- 
ent variable  undergoes  a  given  change  of  value.  The  inde- 
pendent variable  is  usually  a  straight  line  or  an  angle,  varying 
between  known  limits ;  the  required  magnitude  is  either  a 
line  regarded  as  generated  by  the  motion  of  a  point,  an  area 
generated  by  the  motion  of  a  line,  or  a  solid  generated  by  the 
motion  of  an  area.  A  plane  area  may  be  generated  by  the 
motion  of  a  straight  line,  generally  of  variable  length,  the 
method  selected  depending  upon  the  mode  in  which  the 
boundaries  of  the  area  are  defined. 


An  Area  Generated  by  a  Variable  Line  having  a  Fixed 

Direction, 

103.  The  differential  of  the  area  generated  by  the  ordinate 
of  a  curve,  whose  equation  is  given  in  rectangular  coordinates, 
has  been  derived  in  Art.  3.  The  same  method  may  be  em- 
ployed in  the  case  of  any  area  generated  by  a  straight  line 
whose  direction  is  invariable. 


124  GEOMETRICAL  APPLICATIONS.  [Art.  I03. 


Let  AB  be  the  generating  line,  and  let  R  be  its  intersection 
with  a  fixed  line  CD^  to  which  it  is  always 
perpendicular.  Suppose  R  to  move  uni- 
formly along  CDy  and  let  RS  be  the  space 
described  by  R  in  the  interval  of  time,  dt. 

U D     Then   the  value   of  the    differential    of  the 

!  area,  at  the  instant  when  the  generating  line 

passes  the  position  AB^  is  the  area  which 
would  be  generated  in  the  time  dt^  if  the 
rate  of  the  area  were  constant.  This  rate 
would  evidently  become  constant  if  the  generating  line  were 
made  constant  in  length  ;  and  therefore  the  differential  is  the 
rectangle,  represented  in  the  figure,  whose  base  and  altitude 
are  AB  and  RS ;  that  is,  it  is  the  product  of  the  generating  line^ 
and  the  differential  of  its  motion  in  a  direction  perpendicular  to 
its  length. 

104.  In  the  algebraic  expression  of  this  principle,  the  inde- 
pendent variable  is  the  distance  of  R  from  some  fixed  origin 
upon  CD,  and  the  length  of  AB  is  to  be  expressed  in  terms 
of  this  independent  variable. 

When  the  curve  or  curves  defining  the  length  of  AB  are 
given  in  rectangular  coordinates,  CD  is  generally  one  of  the 
axes;  thus,  if  the  generating  line  is  the  ordinate  of  a  curve, 
the  differential  is  y  dx,  as  shown  in  Art.  3.  It  is  often,  how- 
ever, convenient  to  regard  the  area  as  generated  by  some 
other  line. 

For  example,  given  the  curve  known  as  the  witch,  whose 
equation  is 

^ X  —  2af' -\- ^X  r=.  o (i) 

This  curve  passes  through  the  origin,  is  symmetrical  to  the 
axis  of  X,  and  has  the  line  x  =  2a  for  an  asymptote,  since 
X  =  2a  makes  y  =  ±  00  . 

Let  the  area  between  the  curve  and  its  asymptote  be  re- 


\ 


§  VIII.]    AREAS  GENERATED  BY    VARIABLE  LINES. 


125 


quired.     We  may  regard  this  area  as  generated  by  the  line 
PQ  parallel  to  the  axis  of  ;r,  y  being  taken 
as  the  independent  variable.     Now 


PQ  =  2a  —  Xy 
hence  the  required  area  is 

A  =  \      {2a-  x)dy .     .     .     .     (2) 

From  the  equation  (i)   of  the  curve,  we 
have 

_      2a^ 


whence        2a  —  x 


M 


Fig.  4. 


and  equation  (2)  becomes 

^  =  8^r     ^^^    -  =  4/y^tan-^J^T    =4;r^^ 

J_«,/+4^?2       ^  2tf  L« 


Oblique  Coordinates. 

(05.  When  the  coordinate  axes  are  oblique,  if  a  denotes 
the  angle  between  them,  and  the  ordinate  is  the  generating 
line,  the  differential  of  its  motion  in  a  direction  perpendicular 
to  its  length  is  evidently  sin  a-dx ;  therefore,  the  expression 
for  the  area  is 


^  =  sin  «f  M/  dx. 


126  GEOMETRICAL  APPLICATIONS.  [Art.  I05. 

As  an  illustration  let  the  area  between  a  parabola  and  a  chord 
passing  through  the  focus  be  required.  It  is  shown  in  treatises 
on  conic  sections,  the  expression  for  a  focal  chord  is 


AB  —  \a(iosQ(?a  ^      .      .     .      (i) 

X 

where  a  is  the  inclination  of  the  chord 
to  the  axis  of  the  curve,  and  a  is  the 
distance  from  the  focus  to  the  vertex. 
It  is  also  shown  that  the  equation  of 
the  curve  referred  to  the  diameter 
which  bisects  the  chord,  and  the  tan- 
gent at  its  extremity  which  is  parallel  to  the  chord  is 

j^  —  4^  cosec^  a-x (2) 

The  required  area  may  be  generated  by  the  double  ordi- 
nate in  this  equation;  and  since  from  (i)  the  final  value  of 
J/  is  ±  2^  cosec^  oc,  equation  (2)  gives  for  the  final  value  of  x 

OR  =  a  cosec^  a. 
Hence  we  have 


Fig.  5. 


(a  cosec^a 

y4  =  2  sin  «f  y  dx^ 

J  o 

or  by  equation  (2) 

(a  cosec^a 
\/xdx  = 
o 


Sa^  cosec^  a 


3 

Employment  of  an  Auxiliary    Variable, 
106.  We  have  hitherto  assumed  that,  in  the  expression 


A 


ydx, 


§VIII.]     EMPLOYMENT  OF  AN  AUXILIARY  VARIABLE.        12J 

X  is  taken  as  the  independent  variable,  so  that  dx  may  be 
assumed  constant ;  and  it  is  usual  to  take  the  limits  in  such  a 
manner  that  dx  is  positive.  The  resulting  value  of  A  will 
then  have  the  sign  of  j,  and  will  change  sign  if  y  changes 
sign. 

It  is  frequently  desirable,  however,  as  in  the  illustration 
given  below,  to  express  both  y  and  dx  in  terms  of  some  other 
variable.  When  this  is  done,  it  is  to  be  noticed  that  it  is  not 
necessary  that  dx  should  retain  the  same  sign  throughout  the 
entire  integral.  The  limits  may  often  be  so  taken  that  the  ex- 
tremityof  the  generating  ordinate  must  pass  completely  around 
a  closed  curve,  and  in  that  case  it  is  easily  seen  that  the  com- 
plete integral,  which  represents  the  algebraic  sum  of  the  areas 
generated  positively  and  negatively,  will  be  the  whole  area  of 
the  closed  curve. 

107.  As  an  illustration,  let  the  whole  area  of  the  closed 
curve 


f 


I, 


©'  *  (f) 

represented  in  Fig.  6,  be  required.     If  in  this  equation  we  put 


we  shall  have 


©'= 


cos  tp ; 


whence  ^  =  ^  sin^ //',  and  y  =  b  cos^  ip ,     .     .     (i 

Therefore  \y  dx  =  ^ab  cos^  ip  sin^  ^  dip. 


128 


GEOMETRICAL  APPLICATIONS, 


[Art.  I07. 


Now  if  in  this  integral  we  use  the  Hmits  o  and  27r,  the  point 

determined  by  equation  (i)  de- 
scribes the  whole  curve  in  the 
direction  A  BCD  A.  Hence  we 
have  for  the  whole  area 


(277 
cos^  ^  sin^  ^  di\)^ 

and  by  formula  (0 
3-I-I        _  '^^nab 


^      6-4-2  8 


The  areas  in  this  case  are  all  generated  with  the  positive 
sign,  since  when  j/  is  negative  dx  is  also  negative.  Had  the 
generating  point  moved  about  the  curve  in  the  opposite  direc- 
tion, the  result  would  have  been  negative. 


Area  generated  by  a  Rotating  Line  or  Radius  Vector. 


(08.  The  radius  vector  of  a  curve  given  in  polar  coordinates  is 
a  variable  line  rotating  about  a  fixed  extremity.     The  angular 

rate  is  denoted  by  ^    and   may 


dt 


be    re- 


garded as  constant,  although  the  rate  at 
which  area  is  generated  by  the  radius 
vector  OPy  Fig.  7,  is  not  constant,  be- 
cause the  length  of  OP  is  not  constant. 
The  differential  of  this  area  is  the 
area  which  would  be  generated  in  the 
time  dt,  if  the  rate  of  the  area  were  con- 
stant ;    that  is  to  say,  if  the 


Fig.  7. 
radius  vector  were  of  constant 


VIII.]       AREAS  GENERATED  BY  ROTATING  LINES.  1 29 


length.     It  is  therefore  the  circular  sector  OPR  of  which  the 
radius  is  r  and  the  angle  at  the  centre  is  dd. 


Since 


arc  PR  =  r  dd, 


sector  OPR^-r"  dd\ 

2  ' 


therefore  the  expression  for  the  generated  area  is 


(I) 


109.  As  an  illustration,  let  us 
find  the  area  of  the  right-hand  loop 
of  the  lemniscata 

7^=  a^  cos  26. 


Fig.  8. 


The  limits  to  be  employed  are   those    values  of    6  which 

make  r  =  o ;  that  is and  -. 

4  4 

Hence  the  area  of  the  loop  is 


-=?/:= 


9 

COS  20  dd  =  -  sin  26 
4 


110.  When  the  radii  vectores,  r^  and  r^  corresponding  to  the 
same  value  of  6  in  two  curves,  have  the  same  sign,  the  area 
generated  by  their  difference  is  the  difference  of  the  polar  areas 
generated  by  r^  and  r^.     Hence  the  expression  for  this  area  is 


\ 


=ii<'.' 


ri')  dd. 


(2) 


I30 


GEOMETRICAL   APPLICATIONS. 


[Art.  III. 


111.  Let   us    apply  this  formula    to    find    the   whole    area 
between  the  cissoid 


Tx  =■  2a  (sec  B  —  cos  6), 

Fig.  9,  and  its  asymptote  BP2y  whose 
polar  equation  is 

^2  =  2a  sec  0. 

One  half  of  the  required  area  is  generated 
by  the  line  PtP2,  while  6  varies  from  o  to 
I 


TT.     Hence  by  the  formula 


Fig.  9. 


A  =  2^2 J^'  (2-cos2^)  d6  =  ^7ra\ 
Therefore  the  whole  area  required  is  ^Tta^. 


Transformation  of  the  Polar  Formulas, 

112.  In  the  case  of  curves  given  in  rectangular  coordinates, 
it  is  sometimes  convenient  to  regard  an  area  as  generated  by  a 
radius  vector,  and  to  use  the  transformations  deduced  below 
in  place  of  the  polar  formulas. 


Put 


y  =  fnx ; 


(I) 


now  taking  the  origin  as  pole  and  the  initial  line  as  the  axis 
of  ,r,  we  have 

X  =  r  cos  6, 


therefore 
and 


y  —  r  ^\ViQ\     .     .     •     (2) 
==^=tan^, 

X 

dm  =  sec^  0  dd (3) 


m 


§  VI 1 1.]     TRANSFORMA  TION  OF  THE  POLAR  FORMULA  S.      1 3 1 
From  equations  (2)  and  (3), 

j(^  dm  =  r^  dO  ; 

therefore  equation  (i)  of  Art.  108  gives 

A  =  —  Li'^  dm.      ......      (4) 

In  like  manner,  equation  (2)  Art.  1 10  becomes 

A^\\{xi-x?)dm (5) 

(13.  As  an  illustration,  let  us  take  the  folium      ^ 


;r^  +  y  —  3«;rj/ =  O (i) 

Putting  y  =  mx^  we  have 

:t^  ( I  -\-  if^)  —  lamx^  —o (2) 

This  equation  gives  three  roots  or  values  of  ;r,  of  which  two 
are  always  equal  zero,  and  the  third  is 


;r  =  J^; <4) 


whence  _    ^am ,. 

These  are  therefore  the  coordinates  of  the  point  P  in  Fig.  10.  o 
Since  the  values  of  x  and  y  vanish  when  m  =  o,  and  when  / 
m  =  c^ ,  the  curve  has  a  loop  in  the  first  quadrant.     To  find      \ 


132 


GEO  ME  TRIG  A  L  A  PPLIGA  TIONS. 


[Art.  113. 


the  area  of  this  loop  we  therefore  have,  by  equation  (4)  of  the 
preceding  article, 


2 


n^  dm 


Jo(i  + 


(i  +  n^f 


3^ 
2 


I  +  m^A^         2 


114.  The  area  included  between  this  curve  and  its  asymp- 
p      tote  may  be  found  by  means  of  equation 
(5),  Art.  112.     The  equation  of  a  straight 
line  is  of  the  form 


D\   0 
c 


y  =  mx  +  by 


Fig.  10. 


and  since  this  line  is  parallel  to  ^  =  mx^ 
the  value  of  m  for  the  asymptote  must  be 

that  which  makes  x  and  y  in  equations  (4)  and  (5)  infinite ; 

that  is,  //^  =  —  I  ;  hence  the  equation  of  the  asymptote  is 


y  Ar  X  —  b, 


(6) 


in  which  b  is  to  be  determined.  Since  when  m=  —  i,  the 
point  P  of  the  curve  approaches  indefinitely  near  to  the  asymp- 
tote, equation  (6)  must  be  satisfied  by  P  when  m=  —  i. 
From  (4)  and  (5)  we  derive 

m^  4-  m             xain 
y  -\-  X  =  ^a  — ■ -„  = ^ r  ; 

whence,  putting  m  =  —  i,  and  substituting  in  equation  (6) 

—  a  =  by 
the  equation  of  the  asymptote  AB,  Fig.  10,  is 

y  -\-  x=  -a c     (7) 


§  VI 1 1 .]     TRANSFORMA  TION  OF  THE  POLAR  FORMULA  S*      1 3  3 

Now,  as  m  varies  from  —  oo  to  o,  the  difference  between  the 
radii  vectores  of  the  asymptote  and  curve  will  generate  the 
areas  OBC  and  ODA,  hence  the  sum  of  these  areas  is  repre- 
sented by 

A  =  -\     {xi  —  x^)  dm, 


lly-- 


in  which  Xc^  is  taken  from  the  equation  of  the  asymptote  (7), 
and  Xy_  from  that  of  the  curve. 
Putting  J  =  mx,  in  (7),  we  have 


a 

X2=   - 


I  +  m 


and  the  value  of  x^  is  given  in  equation  (4).     Hence 

^  ^  r    3 L_T 


1  +  ;;r 


m 


2    I  ~  7n  +  m 


G 


o  4f 


^. 


Adding  the  triangle  OCD,  whose  area  is  \a^,  we  have  for  the 
whole  area  required  ^d^. 


*  This  reduction  is  given  to  show  that  the  integral  is  not  infinite  for  the 
value  m=  —  I,  which  is  between  the  limits.     See  Art.  77. 


134  GEOMETRICAL  APPLICATIONS.  [Ex.  VIII. 


Examples  VIII. 

I.  Find  the  area  included  between  the  curve 

/d  the  axis  of  x, 
2.  Find  the  whole  area  of  the  curve 

a^y^  ■-  x^  (a-  —  ^'). 
\J    3.  Find  the  area  of  a  loop  of  the  curve 


13 


J 


4.  Find  the  area  between  the  axes  and  the  curve 

y{x'  ■ha'')=d'{a-x).  ^^  p  -  ^^1 

5.  Find  the  area  between  the  curve 


22.      22         22 
xy  +  ay   —  a  x^  =  o, 


and  one  of  its  asymptotes.  '  2«*. 

^  /        6.  Find  the  area  between  the  parabola  y  =  ^ax  and  the  straight 

line  y  =  X.  ^,  — , 

3 

7.  Find  the  area  of  the  ellipse  whose  equation  is 

ax^  +  2bxy  +  cy'  =  i.  ^,(J- i'y 


/ 

§VIIL]  EXAMPLES.  135 

i-^ .. . — _ _ 

\y       8.  Find  the  area  of  the  loop  of  the  curve 
cy"  =  (x  —  d){x  —  by  J 
in  which  ^  >  o  and  b  y  a. 


8  (^  -  a)% 


\/     9.  Find  the  area  of  the  loop  of  the  curve 

ay  =  X*  (b  +  x). 
10.  Find  the  area  included  between  the  axes  and  the  curve 


105^8 


=i       /  V  \  ^  ab 

I.  — 


(:-)"-©• 


20 


\.    II.  If  «  is  an  integer,  prove  that  the  area  included  between  the 
axes  and  the  curve 


&-&  = 


.              n(n—  1)  '  '  '  1  , 

IS  A  =  — r-^ — - — ; — c  ab. 


12.  If  n'ls  an  odd  integer,  prove  that  the  area  included  between 
the  axes  and  the  curve 


\n(n—  2)  '♦'!]'  nab 


IS  A  —        .  . 

271  \2n  —  2)  •  •  •   2        2 


136 


GEOMETRICAL  APPLICATIONS.  [Ex.  VIII. 


13.  In  the  case  of  the  curtate  cycloid 

X  =  aip  —  b  ?>m  tj.^  y  =  a  —  b  cos  ^, 

find  the  area  between  the  axis  of  x  and  the  arc  below  this  axis. 


(2a'  +  b")  cos-^l  -  za  Vib'  -  «"). 


14.  li  b—  iaTTj  show   that  the   area   of  the  loop  of  the  curtate 
cycloid  is 

15.  Find  the  area  of  the  segment  of  the  hyperbola 
^j^-^      >  X  =  a  sec  ip,  y  =  b  tan  t/j^ 

cut  off  by  the  double  ordinate  whose  length  is  2b. 


ab 


V2 


16.  Find  the  whole  area  of  the  curve 

r""  —  a"  cos'  0  ■¥  b""  sin"  9. 

17.  Find  the  area  of  a  loop  of  the  curve 

r"  =  d'  cos'  0  —  ^'  sin''  0. 


log  tan  ^] 


2  ^ 


ab       (a'-b')        _,a 

—  +  ^^ ^  tan      - 

22  b 


\^        18.  Find  the  areas  of  the  large  and  of  each  of  the  small  loops  of 
the  curve 

r  —  a  cos  Q  cos  2O  : 


§  VI 1 1.]  EXAMPLES,  137 

and  show  that  the  sum  of  the  loops  may  be  expressed  by  a  single 
integral. 


s/. 


nc^   .  a  .       7ta       a 

-J-  +  -  ,     and . 

16       4  '  32       8 


12 


9.  In  the  case  of  the  spiral  of  Archimedes,   ^  3  ^        ^ 

find  the  area  generated  by  the  radius  vector  of  the  first  whorl  and 
that  generated  by  the  difference  between  the  radii  vectores  of  the  «th 
and  (n  +  i)th  whorl. 

^,      and      8««V.-''^'« 
6 

20.  Find  the  area  of  a  loop  of  the  curve 
r  =  a  sm  39. 

21.  Find  the  area  of  the  cardioid 
r  =  4a  sin'  ^.  6;r«'. 

22.  Find  the  area  of  the  loop  of  the  curve 

cos  29  a^  (4  —  Tt) 

r  =  a^^ — -. -. 

cos  9  2 

23.  In  the  case  of  the  hyperbolic  spiral, 

rB  =  ay 

show  that  the  area  generated  by  the  radius  vector  is  proportional  to 
the  difference  between  its  initial  and  its  final  value. 


138  GEOMETRICAL  APPLICATIONS.  [Ex.  VI I L 

24.  Find  the  area  of  a  loop  of  the  curve 

r^=^  a  cos  n  9.  . 

25.  Find  the  area  of  a  loop  of  the  curve 

„  _    .  sin  38  •  d" 

7     —  u  a  fl   •  — • 

COS   o  2 

26.  Find  the  area  of  a  loop  of  the  curve 

r'  sin  0  =  <3!*  cos  28. 

Notice  that  r  z>  ?ra/  and  finite  from  0  =  ^  to^  =  —  ,  and  that   — — 
•^         •'  4  4  '  J  sm  Q 

is  negative  in  this  interval.  d\    ^2  ~  log  (i  +   ^2)     . 

»    /   27.  Find  the  area  of  a  loop  of  the  curve  X 

(x'^yy^a^xy. 

Transform  to  polar  coordinates.  — . 

28.  In  the  case  of  the  lima9on 

r  =  2a  cos  B  +  If, 

find  the  whole  area  of  the  curve  when  b>  2a  and  show  that  the  same 
expression  gives  the  sum  of  the  loops  when  /'  <  2a. 

(2a^  +  ^');r. 


§  VIII.]  EXAMPLES,  139 

29.  Find  separately  the  areas  of  the  large  and  small  loops  of  the 
lima9on  when  b  <  2a. 

If  o'  =  cos-M  —  —  ) , 

large  loop  =  (2^'  +  h')  a -V  ^  V(4«'  -  b'') ; 
small  loop  =  {^2d'  +  b"")  {n  —  a)  —  ^  ^(4^^  -  b'\ 


30.  Find  the  area  of  a  loop  of  the  curve 

/-'  r=  d^  cos  n^  +  /$^  sin  «  9. 

31.  Find  the  area  of  the  loop  of  the  curve 

2  cos  2  6'  —  I 


\/{a'  +  b') 


r  =  a 


cos  0 


[5V3-|->. 


32.   Show  that  the  sectorial  area  between  the  axis  of  x^  the  equi- 
lateral hyperbola 

-r'  -/  =  I, 

and  the  radius  vector  making  the  angle  6  at  the  centre  is  represented 
by  the  formula 

.        I  -       I  +  tan  0 

^  =  -  log ; 

4      ^  I  —  tan  Q  ' 

and  hence  show  that 

f2A    _|_     ^  -  2A  g2A   f-2A 

.V  = ,  and  y  =  . 

2  2 

If  A  denotes  the  corresponding  area  in  the  case  of  the  circle 

x'  +/  =  I, 
we  have 

X  =  cos  2^,  and  y  =  sin  2^. 


HO  GEOMETRICAL  APPLICATIONS.  [Ex.  VII L 

In  accordance  with  the  analogy  thus  presented^  the  values  of  x  and  y  given 
above  are  called  the  hyperbolic  cosine  and  the  hyperbolic  sine  of  2  A.      Thus 

f2A    ^_     ^-2A  ^_2A f2A 

—  :=  cosh  (2^), —  sinh  (2A). 


\^     33.  Find  the  area  of  the  loop  of  the  curve 

JI-*  ^"  3«^y  +  2«y  =  o.    ^^ 

34.  Find  the  area  of  the   oop  .of  the  curve 


.  -^    s'^' 


\    38.  Trace  the  curve 


•  y 

AT  =  2^  sin  — , 

X 


35" 


lyfj    -f-     T 

;»;2«  +  i4.jj,2«  +  i  —  (2«+  i)  «Jt:«_y«.  - — a^  , 


35.  Find  the  area  between  the  curve 

jp2«  +  i  _|_^2«+i  —  (^2n  +  i)  axy*" 

.    .  2«   +   I      „ 

and  its  asymptote.  a  . 

36.  Find  the  area  of  the  loop  of  the  curve 

y^  +  ax^  —  axy  =  o. 

37.  Find  the  area  of  a  loop  of  the  curve 

x^  4-  jj;*  =  c^xy. 


60 


and  find  the  area  of  one  loop. 


§IX.] 


VOLUMES   OF  GEOMETRIC  SOLIDS. 


141 


IX.  J^'^~ 

Volumes  of  Geometric  Solids. 


x^ 


115.  A  geometric  solid  whose  volume  is  required  is  fre- 
quently defined  in  such  a  way  that  the  area  of  the  plane  sec- 
tion parallel  to  a  fixed  plane  may  be  expressed  in  terms  of 
the  perpendicular  distance  of  the  section  from  the  fixed  plane. 
When  this  is  the  case,  the  solid  is  to  be  regarded  as  generated 
by  the  motion  of  the  plane  section,  and  its  differential,  when 
thus  considered,  is  readily  expressed. 

116.  For  example,  let  us  consider  the  solid  whose  surface  is 
formed  by  the  revolution  of  the  curve  APB^  Fig.  ii,  about 
the  axis  OX.  The  plane  section  per- 
pendicular to  the  axis  OX  \s  a  circle; 
and  if  APB  be  referred  to  rectangu- 
lar coordinates,  the  distance  of  the 
section  from  a  parallel  plane  passing- 
through  the  origin  is  jr,  while  the 
radius  of  the  circle  \s  y.  Supposing 
the  centre  of  the  section  to  move 
uniformly  along  the  axis,  the  rate  at 
which  the  volume  is  generated  is  not 
uniform,  but  its  differential  is  the  vol- 
ume which  would  be  generated  While  the  centre  is  describing 
the  distance  dx,  if  the  rate  were  made  constant.  This  differen- 
tial volume  is  therefore  the  cylinder  whose  altitude  is  dx,  and 
the  radius  of  whose  base  isj^.     Hence,  if  F  denote  the  volume, 

dV  =  Tty^  dx. 

117.  As  an  illustration,  let  it  be  required  to  find  the  volume 
of  the  paraboloid,  whose  height  is  h^  and  the  radius  of  whose 
base  is  b. 


142  GEOMETRICAL  APPLICATIONS.  [Art.  1 1 7. 

The  revolving  curve  is  in  this  case  a  parabola,  v/hose  equa- 
tion is  of  the  form 

and  since  y  =  b  when  x  —  h, 

U^  —  ^ky  whence  4<^  =r  -7  ; 

ft 

the  equation  of  the  parabola  is  therefore 


,      ^ 
f  =  -^.. 


Hence  the  volume  required  is 


y—  TV  \    y"^ dx  =  n  —\    x dx 


nb'h 
2 


1(8.  It  can  obviously  be  shown,  by  the  method  used  in 
Art.  116,  that  whatever  be  the  shape  of  the  section  parallel  to 
a  fixed  plane,  the  differential  of  the  volume  is  the  product  of  the 
area  of  the  generating  section  and  the  differential  of  its  ^notion 
perpendicular  to  its  plane. 

If  the  volume  is  completely  enclosed  by  a  surface  whose 
equation  is  given  in  the  rectangular  coordinates  x,  y,  2,  and  if 
we  denote  the  areas  of  the  sections  perpendicular  to  the  axes 
by  Ajr,  Ay,  and  A^.,  we  may  employ  either  of  the  formulas 

V  =  \a^.  dx,  V ^  [a,,  dy,  V=:\a,  dz. 

The  equation  of  the  section  perpendicular  to  the  axis  of  x 
is  determined  by  regarding  x  as  constant  in  the  equation  of 
the  surface,  and  its  area  A^  is  of  course  a  function  of  x. 


§  IX.]  VOLUMES  OF  GEOMETRIC  SOLIDS.  1 43 

For  example,  the  equation  of  the  surface  of  an  ellipsoid  is 

The  section  perpendicular  to  the  axis  of  x  is  the  ellipse 
f      ^  __c?  —  x^ 

b  c 

whose  semi-axes  are-   ^(c^  —  x^)  and  -  V{a^  —  ^. 


Since  the  area  of  an  ellipse  is  the  product  of  tt  and  its  semi- 
axes, 

The  limits  for  x  are  ±a,  the  values  between  which  x  must  lie 
to  make  the  ellipse  possible.     Hence 


nbc  f''    , 


2       .Ji\  ^..  _^7tabc 


x^)  dx 


119.  The  area  A.r  can  frequently  be  determined  by  the  con- 
ditions of  the  problem  without  finding  the  equation  of  tbe 
surface.  For  example,  let  it  be  required  to  find  the  volume  of 
the  solid  generated  by  so  moving  an  ellipse  with  constant 
major  axis,  that  its  center  shall  describe  the  major  axis  of  a 
fixed  ellipse,  to  whose  plane  it  is  perpendicular,  while  the  ex- 
tremities of  its  minor  axis  describe  the  fixed  ellipse.  Let  the 
equation  of  the  fixed  ellipse  be 


144 


GEOMETRICAL  APPLICATIONS. 


[Art.  119. 


and  let  c  be  the  major  semi-axis  of  the  moving  eUipse.  The 
minor  semi-axis  of  this  ellipse  is  y.  Since  the  area  of  an 
ellipse  is  equal  to  it  multiplied  by  the  product  of  its  semi-axes, 
we  have 


Ax=  71  cy  —  —  ^/ic^  —  x^), 
a 


Ttbc  r^ 

Therefore  V= —         V{c^  —  j^)dx\ 

hence,  see  formula  (^), 


F  = 


'^abc 


The  Solid  of  Revolution  regarded  as  Generated  by  a 
Cylindrical  Surface, 

120.  A  solid  of  revolution  may  be   generated   in   another 

manner,  which  is  sometimes  more 
convenient  than  the  employment 
of  a  circular  section,  as  in  Art.  116. 
For  example,  let  the  cissoid  PORy 
Fig.  12,  whose  equation  is 


1 

1 

\ 

J^. 

l^ — 

^ 

A 

■ — 

~c 

p 

v^__ 

\ 

V 


Fig.  12. 


pass  from  the  value  OA 

will  evidently  generate  the  solid  of  revolution 


y^  {2a  —  x)  =  A^f 

revolve  about  its  asymptote  AB. 
The  Hne  PR,  parallel  to  AB  and 
terminated  by  the  curve,  describes 
a  cylindrical  surface.  If  we  con- 
ceive the  radius  of  this  cylinder  to 
2a  to  zero,  the  cylindrical  surface 

Now  every 


§IX.] 


DOUBLE  nVTEGRATIOJV. 


145 


point  of  this  cylindrical  surface  moves  with  a  rate  equal  to 
that  of  the  radius;  therefore  the  differential  of  the  solid  is 
the  product  of  the  cylindrical  surface,  and  the  differential  of 
the  radius.     The  radius  and  altitude  in  this  case  are 


PC=2a 
therefore 

Putting 


x, 


and 


(•za 


2ax 


fxdx. 


.^\i 


X  —  a  =  a  sin  6, 


PR=2y, 


u 


V  =  47ra^   '    (cos^  0  +  cos^  6  sin  6)  dd  =  2;rV. 


Double   Integration, 

121.  When  rectangular  coordinates  are  used,  the  expression 
for  the  area  generated  by  a  line  parallel 
to  the  axis  of  y  and  terminated  by  two 
curves  is 


A^\^^{y^-y^dx. 


(I) 


Let  AB,  in  Fig.   13,  be  the  initial,        o        a  c 

and  CD  the  final  position  of  the  gen-  ^^^-  ^3- 

erating  line,  then  the  area  is  ABDC,  which  is  enclosed  by  the 
curves 

and  by  the  straight  lines 


a, 


x=b. 


14^  GEOMETRICAL  APPLICATIONS.  [Art.  121. 

If  in  equation  (i)  we  substitute  for  y^  —ji  the  equivalent  ex- 
pression    cfyf  we  have 

(2) 


which  expresses  the  area  in  the  form  of  a  double  integral.  In 
this  double  integral  the  limits  ji  and  j/g  forj/,  are  functions  of 
.r,  while  a  and  b^  the  limits  for  x,  are  constants. 

122.  If  the  area  is  that  of  a  closed  curve  y^  and  y2  are  two 
values  of  J/  corresponding  to  the  same  value  of  x  in  the  equa- 
tion of  the  curve,  and  a  and  b  are  the  values  of  x  for  which  y^ 
andjj/g  become  equal,  as  represented  by  the  dotted  lines  in  Fig. 
13.  It  is  evident  that  the  entire  area  may  also  be  expressed  in 
the  form 

A-^\^^yxdy; (3) 

and  that  when  either  of  the  forms  (2)  or  (3)  is  applied  to  the 
area  of  a  closed  curve  the  limits  are  completely  determined  by 
the  equation  of  the  curve. 

123.  The  limits  in  either  of  the  expressions  (i)  or  (2)  define 
a  certain  closed  boundary,  and  since  either  of  these  integrals 
represents  the  included  area,  it  is  evident  that  we  may  write 


II 


dy  dx 


dxdy\ 


provided  it  is  understood  that  the  limits  in  the  two  expressions 
are  such  as  to  represent  the  same  boundary.  It  should  however 
be  noticed  that  if  the  boundary  is  like  that  represented  by  the 
full  lines  in  Fig.  13,  or  if  the  arcs  j/=j/i  and  y  =/2  do  not 
belong  to  the  same  curve,  we  cannot  make  a  practical  application 
of  the  form  (3)  without  breaking  up  t'he  integral  into  several 
parts. 


§IX.] 


DOUBLE  INTEGRATION. 


147 


124.   Let  ^  (-i',j)  be  any  function  of  x  and  j.     In  the  double 
integral 

f    f  '  <i>{x,y)dy  dx, (i) 

J  a  J  J2 


X  is  considered  as  a  constant  or  independent  of  y  in  the  first 
integration,  but  the  limits  of  this  integration  are  functions  of  x. 
The  double  integration  is  then  said  to  exte7id  over  the  area 
which  is  represented  by  the  expression 


f   ^''  dyd,x 


or 


J^  (72-^1) 


dx. 


(2) 


125.  Now  let  the  surface,  of  which 


z  =  ^  (x,  y) 


(3) 


is  the  equation  in  rectangular  coordinates,  be  constructed ;  and 
let  a  cylindrical  surface  be  formed  by  moving  a  line  perpen- 
dicular to  the  plane  of  xy  about  the  boundary  of  the  area  (2) 
over  which  the  integration  extends.  Let  us  suppose  the  value 
of  z  to  be  positive  for  all  values  of  x  and  y  which  represent 
points  within  this  boundary.  Then  the  cylindrical  surface, 
together  with  the  plane  of  xy  and  the  surface  (3),  encloses  a 
solid,  of  which  the  base  is  the  area 
(2)  in  the  plane  xy,  or  ASBR  in  Fig. 
14,  and  the  upper  surface  is  CQDP  a 
2)ortion  of  the  surface  (3). 

Let  SRPQ  be  a  section  of  this 
solid  perpendicular  to  the  axis  of  x. 
In  this  section  x  has  a  constant  value, 
and  the  ordinates  of  R  and  5  are  the 
corresponding  values  of  y^  and  ja- 
The  area  of  this  section,  which  denote  Fig.  14. 


148  GEOMETRICAL  APPLICATIONS.  [Art.  1 25. 

by  A^^  as  in  Art.  1 1 7,  may  be  regarded  as  generated  by  the  line 
z,  hence 


A 
and  therefore 


iy\ 

(b     rja 
\    zdydx (i) 


which  is  identical  with  expression  (i)  Art.  124. 

126.  Now  it  is  evident  that  the  same  volume  may  be  ex- 
pressed by 


=     \zdxdy^ 


provided  that  the  double  integration  extends  over  the  same  area. 
Hence,  with  this  understanding,  we  may  write 


#  (jt, j)  dy  dx  =      (f)  {x, y)  dx dy. 


In  this  formula  x  and  y  may  be  regarded  as  taking  the 
places  of  any  two  variables,  the  limits  of  integration  being 
determined  by  a  given  relation  between  the  variables.  Thus 
we  may  write 

(j)  {u,  v)  dv  du  =  \\(})  {u,  v)  du  dv, 

provided  the  limits  of  integration  are  determined  in  each  case 
by  the  same  relation  between  u  and  v. 

127.  For  example,  if  this  relation  is 

U^  ->r  V^  —  c"^  =.  Qj 


§  IX.]  DOUBLE  INTEGRATION.  149 

the  range  of  values  in  the  first  integration  is  between 
that  is,  we  must  have 

or  1^  ■¥  v^  —  c^  <o (i) 

But  this  condition  also  expresses  the  limits  for  u^  since  v  is 
only  possible  when  u^  <  c^.     Now,  putting   rectangular  coordi- 
nates, X  and  7,  in  place  of  u  and  v,  it  is  convenient  to  express 
the  restriction  (i),  by  saying  that  the  range  of  values  of  x  and    ^ 
y  is  such  as  to  represent  every  point  within  the  circle 

Volumes  by  Double  and  Triple  Integration. 

128.  As  an  application  of  formula  (i).  Art.  125,  let  us  sup-  " 
pose  the  curve  ASBR  to  be  the  circle 

{x-kf  +  {v-kf  =  c\ (I) 

and  the  equation  of  the  surface  CQDP  to  be 

xy^pz (2) 

Then  ^=M'  r  ^ydydx^^J\    {y^-yi)xdx, 

p  J  a  Jj^  ^P  J  a 


150  GEOMETRICAL   APPLICATIONS,  [Art.  1 28. 

in  which  the  limits  y^  and  y^  are   derived   from   equation   (i). 
Hence 

and  ^  =  ^  f '  ^^^  -  (-^  -  ^)']  ^  ^^' 

P   ]  a 

The  limits  for  x  are  the  extreme  values  of  x  which  make  j» 
possible  ;  that  is, 

a  =  h  —  c  and  b  =  h  +  c 

To  evaluate  the  integral,  put 

X  —  h  =  c  sin  6 ; 

then  V=^\\Qos^d{k-\-c  sin  6)  dd. 

Since,  by  Art.  87, 

TT 

cos^  ^  sin  ddd  =G, 
we  have  finally 

~     /      * 

(29.  A  volume  in  general  may  be  represented  by  the  triple 
integral 


V=\\\dgdy,ix, (I) 


§  IX.]  TRIPLE  INTEGRATION.  I5I 


which  is  equivalent  to 

F=||(^2-^i)^J^^, (2) 

for      {zc^  —  ^\)  (iy  —  Axy  the  area  of  a  section  perpendicular  to 

the  axis  of  x.  We  may  regard  this  formula  as  expressing  the 
difference  between  two  cylindrical  solids  of  the  form  represented 
in  Fig.  14. 

130.  When  the  volume  is  that  of  a  closed  surface,  z^  ^"<^  ^1 
are  two  values  of  z  in  terms  of  x  and  j/  found  from  the  equa- 
tion of  the  surface.  The  area  over  which  the  integration 
extends  is  in  this  case  the  projection  of  the  solid  upon  the  plane 
of  xy ;  in  other  words,  the  base  of  a  circumscribing  cylinder. 
Thus,  if  the  volume  is  that  of  the  sphere 

x'  +  f-{-{z-cy=a\ (I) 

^1  and  ^2  ^^^  the  two  values  of  js  derived  from  this  equation* 
that  is  c  ±   */(a^  —  x^  —  j^). 

Hence  ^2  —  ^1  =  2  V{a^  —  x^  —  j/*), 


and 


V=  2  [[  V  (a^  -  ^-f)  dydx..    c     .     .     .  (2) 


The  integration  here  extends  over  the  circle 

x'^f-a^^Q..     .    . (3} 


152  GEOMETRICAL  APPLICATIONS.  [Art.   I30, 

since  z^  —  z^  is  real  only  when 


c?  —  x^  —  f  >  o. 


From  equation  (3)  we  find  the  limits  for  j/  to  be 
hence,  by  formula  {M),  equation  (2)  becomes 


V 


=  n  {c^  —  j^)  dx. 


Finally  the  limits  for  x  are  ±  a^  since  y  is  real  only  when  x  is 
between  these  limits  ; 


therefore  V 


c^x x^  I      —  -  7ta^ , 


Elements  of  Area  and  Volume. 
131.   In  accordance  with  Art.  100,  the  expression  for  an  area, 

J    J     dydx, (i) 


IS  the  limit  of  the  sum 


2li2',\i^y-\c.x. 


Since  each  of  the  terms  included  in  ^^"^  Ay  is  multiplied  by 
the  common  factor  ax,  this  sum  may  be  written  in  the  form 

Sl^^-jA/A^- (2) 


§  IX.]  ELEMENTS  OF  AREA    AND    VOLUME.  1 53 

The  sum  (2)  consists  of  terms  of  the  form 

A7  A;r  ; 

and  this  product  is  called  the  element  of  the  sum  ;  in  like  man- 
ner, the  product 

dy  dx, 

which  takes  the  place  of  t\y  Ax  when  we  pass  to  the  limit  by 
substituting  integration  for  summation,  is  called  the  element  of 
the  integral  (i),  or  of  the  area  represented  by  it. 

132.  We  may  now  regard  the  process  of  double  integration 
as  a  process  of  double  summation,  as  indicated  by  expression 
(2),  followed  by  the  act  of  passing  to  the  limiting  value.  In 
the  first  summation  indicated,  the  elemental  rectangles  corre- 
sponding to  the  same  value  of  x  are  combined  into  the  term 
(^2  ~  y'x)  ^-^'j  which  may  be  called  a  linear  element  of  area,  since 
its  length  is  independent  of  the  symbol  A. 

133.  It  is  easy  to  see  that,  in  a  similar  manner,  when  rec- 
tangular coordinates  are  used,  a  volume  may  be  regarded  as 
the  limiting  value  of  the  sum  of  terms  of  the  form 

A,t'  Ay  Az\ 

and  hence  dx  dy  dz^ 

which  takes  its  place  when  we  pass  to  the  limiting  value  by 
substituting  integration  for  summation,  is  called  the  element  of 
volume. 

If  the  summation  is  effected  in  the  order  ^,  7,  x,  the  first 
operation  combines  the  elements  which  have  common  values 
of  y  and  x  into  the  linear  element  of  volume, 

(Z2-  zi)  AX  Ay, 


154  GEOMETRICAL   APPLICATIONS.  [Art.   1 33. 

The  second  operation  combines  the  linear  elements  correspond- 
ing to  a  common  value  of  x^  over  a  certain  range  of  values  oi  y, 
into  a  term  whose  limiting  value  takes  the  form 

A^  AX. 

This  last  expression  represents  a  lamina  perpendicular  to  the 
axis  of  X,  whose  area  is  A^,  a  section  of  the  solid,  and  whose 
thickness  is  A.r. 


Polar  Elements, 
134.  If  in  the  formula  for  a  polar  area, 

\rl-r^)de, (I) 


A  =  '- 
2  J 


[equation  (2),  Art.  no],  we  substitute  for  -{r^—  rf)  the  equiv- 
r  dr,  we  obtain 

A=\^  [\drde, (2) 

in  which  a  and  ^  are  fixed  limits  for  6. 

Now  it  follows,  from  Art.  126,  that  the  limits  being  deter- 
mined by  a  certain  relation  between  r  and  6,  this  integral  may 
also  be  put  in  the  form 

A=[  r\''de'dr=\\{e^-e^)dr,    ...     (3) 

i  a     h^  J  a 


§  IX.]  POLAR  ELEMENTS.  1 55 

in  which  a  and  b  are  the  limiting  values  of  r,  between  which  6 
is  possible. 

The  expression  r  dr  dOy 

in  equation  (2),  is  called  th.^  polar  element  of  area.* 
135.  The  formula 

A  =  \r{e,-e^dr 


-\r{e,-e,) 


may  also  be  derived  geometrically ;  for  r  (S^  —  6^  is  the  length 
of  an  arc  whose  radius  is  r.  As  r  increases,  this  arc  generates 
the  surface,  and  it  is  plain  that  every  point  has  a  motion, 
whose  differential  is  dry  in  a  direction  perpendicular  to  the  arc. 
136.  In  determining  the  volume  of  a  solid,  it  is  sometimes 
convenient  to  express  ;?  as  a  function  of  the  polar  coordinates 
of  its  projection  in  the  plane  of  xy.  In  this  case  we  employ 
the  linear  element  of  volume, 

(<8'2  —  ^1)  r  dr  ddy 
corresponding  to  the  polar  element  of  area. 

*  It  is  easily  shown  that  the  area  included  between  the  circles  whose  radii  are 
rand  r  +  Ar,  and  the  radii  whose  inclinations  to  the  initial  line  are  6  and  fl  +  A« 
is 

(r+  ^Ar)  Ar  Afl. 

Since  r  4-  i  A  r  is  intermediate  between  r  and  r  +  A  r,  the  limiting  value  of  the 
sum,  of  which  this  is  the  element,  is,  by  Art.  99,  the  integral  of  the  element 

rdrde. 

In  the  summation  corresponding  to  equation  (i),  the  elements  are  first  combined 
into  the  sectorial  element 

while  in  the  summation  corresponding  to  equation  (3),  they  are  first  combined  into 
the  arc-shaped  element 

(r+  iArX^a  -0,)  An 


156 


GEOMETRICAL  APPLICATIONS. 


[Art.  136. 


As  an  illustration,  let  us  determine  the  volume  cut  from  a 
sphere  by  a  right  cylinder,  having  a  radius  of  the  sphere  for 
one  of  its  diameters.  Taking  the  centre  of  the  sphere  as 
the  origin,  the  diameter  of  the  cylinder  as  initial  line,  and  the 
axis  of  z  parallel  to  the  axis  of  the  cylinder,  we  have  for 
every  point  on  the  surface  of  the  sphere 


(I) 


where  a  is  the  radius  of  the  sphere.     Hence 

•^2  —  -S"!  =  2  4/(^2  —  r^). 


and 


V 


-n> 


T^Yrdrde-^ 


L      3 


(^2  -  7^)^ 


dd. 


The  circular  base  passes  through  the  pole,  and  its  equation  is 

r  —  a  cos  6, (2) 

hence  the  limits  for  r  are  o  and  a  cos  6,  and  by  substitution  we 
obtain 


V^—Ui  -sin'6)de. 


The  limits  for  0  are  ±  — ,  the  values  which  make   r  vanish 

in  equation  (2) ;   but  it    is  to  be  noticed   that  the  expression 

(d^  —  r^)t,  for  which  we  have  substituted  <^  sin^  d,  is  always  posi- 
tive^ whereas  sin^  S  is  negative  in  the  fourth  quadrant.  Hence 
the  value  of  V  is  double  the  value  of  the  integral  in  the  first 
quadrant ;  that  is, 


V^^\'  {i-^m^B) 


dd 


27ta'' 
3 


8^ 
9 


§IX.] 


POLAR   ELEMENTS, 


157 


If  a  second  cylinder  whose  diameter  is  the  opposite  radius  of 
the  sphere  be  constructed,  the  whole  volume  removed  from  the 


sphere  is 


^Ttar 


1 6^3 


,  and    the    portion    of   the    sphere    which 


[6^ 


remains  is ,  a  quantity  commensurable  with  the   cube  of 

9. 
the  diameter. 


Polar  Coordinates  in  Space, 


137.  A  point    in    space  may  be  determined   by  the  polar 
coordinates  p,  ^^  and  ^,  of  which  p   de- 
notes the    radius    vector    OP,    Fig.    15, 
^  the  inclination  POR  of  p  to  a  fixed 
plane  passing  through   the  pole,  and   d 
the    angle    ROA,  which  the    projection 
of   p   upon    this    plane    makes   with    a 
fixed  line   in   the   plane.     The  angles  ^ 
and  d  thus  correspond    to  the    latitude 
and  longitude  of  the  point  P  considered 
as  situated  upon  the  surface  of  a  sphere 
whose  radius  is  p.     The  radius  of  the 
circle  of  latitude  BP  is 


Fig.  15. 


PC  =  p  cos  (j>. 

The  motions  of  P,  when  p,  ^,  and  Q  independently  vary,  are 
in  the  directions  of  the  radius  vector  OP  and  of  the  tangents  at 
P  to  the  arcs /7?  and  PB,  The  differentials  of  these  motions 
are  respectively 


dp. 


p  </^,         and         p  cos  <f>  dd ; 


158  GEOMETRICAL  APPLICATIONS.  [Art.   1 37, 

and  since  these  motions  are  mutually  rectangular,  the  element 
of  volume  is  their  product, 

f^  cos  ^  dp  d<l>  dQy 
and  F^fffp^cos^^p^^^^ (i) 


(38a  Performing  the  integration  with  respect  to  /o,  the  for- 
mula becomes 

V^-\\{9l- pl)zosii>d(i>de (2) 

When  the  radius  vector  lies  entirely  within  the  solid,  the  lower 
limit  f\  must  be  taken  equal  zero,  and  we  may  write 

V=-\\p'zo->^d<i>de (3) 

The  element  of  this  double  integral  has  the  form  of  a  pyramid 
with  vertex  at  the  pole. 

If,  on  the  other  hand,  in  formula  (i)  we  perform  first  the 
integration  with  respect  to  ^,  we  have 

F=|j(sin4-sin#i)pVp^^.     ....      (4) 

Taking  the  lower  limit  ^1  =  o,  so  that  the  solid  is  bounded  by 
the  plane  OR  A,  we  have  the  simpler  formula 

V--=Usm(f>fJ'dpdd (5) 

139.  The   formulas  of    the  preceding  article  take  simpler 


§  IX.] 


POLAR   COORDINATES  IN  SPACE. 


159 


forms  when  applied  to  solids  of  revolution.  Let  OZ,  Fig.  15, 
be  the  axis  of  revolution,  then  p  and  6  are  polar  coordinates  of 
the  revolving  curve,  OR  being  the  initial  line.  Now  6  is  in 
this  case  independent  of  p  and  ^,  and  its  limits  are  o  and  27t. 
The  integration  with  reference  to  d  may  therefore  be  performed 
at  once.     Thus  from  (3)  we  obtain 


V 


=t) 


f?  COS  ^  d(l> ; 


(6) 


and  in  each  of  the  formulas  the  factor  27r  may  take  the  place 
of  the  integration  with  reference  to  6. 

140,  As  an  example  of  the  use  of  equation  (6),  let  us  find 
the  volume  generated  by  a  circle  revolving  about  one  of  its 
tangents.  The  initial  line,  being  perpendicular  to  the  axis  of 
revolution,  is  a  diameter ;  hence  if  a  is  the  radius  of  the  circle 
its  equation  is 

ft  —  2a  cos  ^, 


and  the  limits  for  ^  are and  -. 

22 


Substituting  in  (6) 


cos^  (j)  dcj)  —  27i^a^. 


141.  The  following  example  of  the  use  of 
equation  (4),  Art.  138,  is  added  to  illustrate 
the  necessity  of  drawing  a  figure  in,  each 
case  to  determine  the  limits  to  be  employed. 

Let  it  be  required  to  find  the  volume 
generated  by  the  revolution  of  the  cardioid 
about  its  axis,  the  equation  of  the  curve 
being 

p  =  a{i  +sin^),     .     . 


Fig.  16. 


(I) 


l6o  .    GEOMETRICAL   APPLICATIONS.  [Art.  I4I. 

when  the  initial  line  is  perpendicular  to  the  axis  of  the  curve, 
as  in   Fig.  16.      The   figure   shows  that  the  upper  limit  for  ^ 

is  —  ;r,  while  the   lower  limit  is  the  value  of   ^  given  by  equa- 
tion (i) ;  therefore 

sin  (^0  =  I,  and  sin  (j^.  = i. 

a 

The  limits  for  ft  are  evidently  o  and  2a.     Substituting  in  equa- 
tion (4)  Art.  138, 


21.  , 

3        4^Jc 


Examples  IX. 

I.  Find  the  volume  of  the  spheroid  produced  by  the  revolution  of 
the  ellipse, 

about  the  axis  of  x. . 


2.  Find  the  volume  of  a  right  cone  whose  altitude  is  a,  and  the 
radius  of  whose  base  is  d.  nab^ 


3.  Find  the  volume  of  the  solid  produced  by  the  revolution  about 
the  axis  of  x  of  the  area  between  this  axis,  the  cissoid 

y  {2a  —  x)  =  x\ 
and  the  ordinate  of  the  point  («,  a).  S^'tt  (log  2  —  |). 


§  IX.]  EXAMPLES,  l6l 

4.  Find  the  volume  generated  by  the  revolution  of  the  witch, 

y^x  —  2ay^  +  ^d^x  =  o, 
about  its  asymptote. 

See  Art.  104.  dfTt^a^. 

5.  The  equilateral  hyperbola 

x'  —  y^  —  m 

revolves  about  the  axis  of  .v  :  show  that  the  volume  cut  off  by  a  plane 
cutting  the  axis  of  x  perpendicularly  at  a  distance  a  from  the  vertex 
is  equal  to  a  sphere  whose  radius  is  a. 

6.  An  anchor  ring  is  formed  by  the  revolution  of  a  circle  whose 
radius  is  b  about  a  straight  line  in  its  plane  at  a  distance  a  from  its 
centre:  find  its  volume.  i>.  >  0-  2n'^aB\ 

7.  Express  the  volume  of  a  segment  of  a  sphere  in  terms  of  the 
altitude  h  and  the  radii  a^  and  a^  of  the  bases. 

—  {le  +  3^r  -I-  3«/). 

8.  Find  the  volume  generated  by  the  revolution  of  the  cycloid, 

x  =  a(ip  —  'SAiiip),  y  =  a  {i  —  cos ^■), 

about  its  base.  57^''^^ 

9.  The  area  included  between  the  cycloid  and  tangents  at  the 
cusp  and  at  the  vertex  revolves  about  the  latter  ;  find  the  volume  gen- 
erated. 

10.  Find  the  volume  generated  by  the  revolution  of  the  part  of  the 
curve 

y=8-, 

which  is  on  the  left  of  the  origin,  about  the  axis  of  x. 

7t 


1 62  GEOMETRICAL   APPLICATIONS.  [Ex.  IX 

11.  The  axes  of  two  equal  right  circular  cylinders,  whose  common 
radius  is  a,  intersect  at  the  angle  a  ;  find  the  volume  common  to  the 
cylinders. 

The  section  parallel  to  the  axes  is  a  rhombus.  i6a^ 

3  sin  «  * 

12.  Find  the  volume  generated  by  the  revolution  of  one  branch  of 
the  sinusoid, 

V  =  ^  sin  — , 
a 

about  the  axis  of  x.  n'^b  ^ 

2a  ' 


13.  Find  the  volume  enclosed  by  the  surface  generated  by  the  revo- 
lution of  an  arc  of  a  parabola  about  a  chord,  whose  length  is  2e,  per- 
pendicular to  the  axis,  and  at  a  distance  b  from  the  vertex. 

i67r^V 

15 

14.  Find  the  volume  generated  by  the  revolution  of  the  tractrix, 
whose  differential  equation  is 


dx        -^  V{a-/) 
about  the  axis  of  x. 

Express  ny^  dx  in  terms  of  y. 


15.  Find  the  volume  cut  from  a  right  circular  cylinder  whose  radius 
is  «,  by  a  plane  passing  through  the  centre  of  the  base,  and  making 
the  angle  a  with  the  plane  of  the  base. 


16.  Find  the  volume  generated  by  the  curve 
xy^  =  ^a^  (2a  —  x) 
revolving  about  its  asymptote.  47rV 


J  ^8 


§  IX.]  EXAMPLES.  163 

•J       17.  Express   the  volume  of    a  frustum  of  a  cone    in  terms  of   its 

height  hy  and  the  radii  ax  and  a^.  of  its  bases. 

7th  .    ^  «x 

—  (^1  +  ^1^0  +  a:), 
3 

18.  Find  the  volume  generated  by  the  revolution  of  the  cardioid, 

r  =  ^  (i  —  cos  9), 
about  the  initial  line. 

Express  y  and  dx  in  terms  of  B.  8  rra^ 

3 

19.  Find  the  volume  of  a  barrel  whose  height  is  2h,  and  diameter 
2b,  the  longitudinal  section  through  the  centre  being  a  segment  of  an 
ellipse  whose  foci  are  in  the  ends  of  the  barrel. 

2h'  +  s^'^ 


27tb'^h 


3  (^^  +  ^0  • 


20.  Find  the  volume  generated  by  the  superior  and  by  the  inferior 
branch  of  the  conchoid  each  revolving  about  the  directrix ;  the 
equation,  when  the  axis  oi  y  is  the  directrix,  being 


xy  =  (^  +  xy  {b'  -  x"). 

n'ab'  ±  ^ 
3 


2   ,.    ,    4^^' 
Tt^ab  ± 


21.  On  two  opposite  lateral  faces  of  a  rectangular  parallelopiped 
whose  base  is  ab,  oblique  lines  are  drawn,  cutting  off  the  distances 
Ci,  Ci,  ^3,  Ci  on  the  lateral  edges.  A  straight  line  intersecting  each  of 
these  lines  moves  across  the  parallelopiped,  remaining  always  parallel 
to  the  other  lateral  faces  :  find  the  volume  cut  off. 

ab  {ci  +  r.  +  ^8  +  ^i) 

4  *    . 

22.  Find  the  volume  enclosed  by  the  surface  generated  by  an  arc 
of  a  circle  whose  radius  is  a,  about  a  chord  whose  length  is  2c. 

3  " 


1 64  GEO  ME  TRICAL  AP  PLICA  TJONS.  [Ex.  IX. 


23.  The  area  included  between  a  quadrant  of  the  ellipse 

A  —  a  cos  ^,  y  =  b  sm  (j), 

and  the  tangents  at  its  extremities  revolves  about  the  tangent  at  the 
extremity  of  the  minor  axis  ;  find  the  volume  generated. 

Ttab"^  (10  —  3  tt) 

24.  An  ellipse  revolves  about  the  tangent  at  the  extremity  of  its 
major  axis  ;  express  the  entire  volume  in  the  form  of  an  integral, 
whose  limits  are  o  and  2  7r,  and  find  its  value.  211^ d^b. 

25.  Show  that  the  volume  between  the  surface, 

s«  =::  a\x''  -^  b\y\ 

and  any  plane  parallel  to  the  plane  of  xy  is  equal  to  the  circumscrib- 
ing cylinder  divided  by  n  ■\-  1. 

26.  A  straight  line  of  fixed  length  2c  moves  with  its  extremities  in 
two  fixed  perpendicular  straight  lines  not  in  the  same  plane,  and  at  a 
distance  2b.  Prove  that  every  point  in  the  moving  line  describes  an 
ellipse  in  a  plane  parallel  to  both  the  fixed  lines,  and  find  the  volume 
enclosed  by  the  generated  surface.  ^n.  {c^  —  Ir)  b 

3 

27.  Find  the  volume  enclosed  by  the  surface  whose  equation  is 

x^       y^       s*  Znabc 

a        b'       c'  5 

28.  A  moving  straight  line,  which  is  always  perpendicular  to  a  fixed 
straight  line  through  which  it  passes,  passes  also  through  the  circum- 
ference of  a  circle  whose  radius  is  a,  in  a  plane  parallel  to  the  fixed 
straight  line  and  at  a  distance  b  from  it  ;  find  the  volume  enclosed 
by  the  surface  generated  and  the  circle.  na^b 


§  IX.]  EXAMPLES.  165 

29.  Find  the  volume  enclosed  by  the  surface 


oca 

and  the  plane  a  .=  a. 

30.  Find  the  volume  enclosed  by  the  surface 

a.  2.  s  V 

x-^  -f-  V*  4-  s^  =  rt!-\ 


Ttabc 
2 


Find  Az  as  in  Art.  107,  and  then  evaluate  V  by  a  similar  method. 

^Tta^ 

31.  Find  the  volume  between  the  coordinate  planes  and  the  surface 


(^,(^U^=. 


\a  J  \bj         \c/  go 

32.  Find  the  volume  cut  from  the  paraboloid  of  revolution 
J*  -f  s'  =  4ax 
by  the  right  circular  cylinder 

^"  +y  =  2ax, 

whose  axis  intersects  the  axis  of  the  paraboloid  perpendicularly  at  the 

focus,  and  whose  surface  passes  through  the  vertex.  ,        i6a^ 

27ta^  + .     • 

3 
7,2i-  The  paraboloid  of  revolution 

.v'^  +  /  =  cz 

is  pierced  by  the  right  circular  cylinder 

x^  -^r  y^  =  ax^ 


1 66  GEOMETKICAL  APPLICATIONS.  [Ex.  IX. 

whose  diameter  is  a,  and  whose  surface  contains  the  axis  of  the  parab- 
oloid ;  find  the  volume  between  the  plane  of  xy  and  the  surfaces  of 
the  paraboloid  and  of  the  cylinder.  2>'^a^ 

34.  Find  the  volume  cut  from  a  sphere  whose  radius  is  ^  by  a 
right  circular  cylinder  whose  radius  is  h^  and  whose  axis  passes  through 
the  centre  of  the  sphere.  A'^V  -^       ,   1       zqx J~l 

_^,-_(^_^)  J. 

35.  Find  the  volume  cut  from  a  sphere  whose  radius  is  a  by  the 
cylinder  whose  base  is  the  curve 

r  =  « cos  39.  2a^7t        8^^ 

3  9 

-^d.  Find  the  volume  cut  from  a  sphere  whose  radius  is  a  by  the 
cylinder  whose  base  is  the  curve 

r  =  a  cos  B  -\-  0   sm  G, 

z  ^                                                          47r^^       16  ,  2       72x4 
supposmg  b  <  a. {a  —  0  y  . 

•J  y 

37.  A  right  cone,  the  radius  of  whose  base  is  a  and  whose  alti- 
tude is  d,  is  pierced  by  a  cylinder  whose  base  is  a  circle  having  for 
diameter  a  radius  of  the  base  of  the  cone  ;  find  the  volume  common 
to  the  cone  and  the  cylinder.  da^ ,  ^ , 

-(9^-16). 

38.  The  axis  of  a  right  cone  whose  semi- vertical  angle  is  a  coin- 
cides with  a  diameter  of  the  sphere  whose  radius  is  a,  the  vertex  being 
on  the  surface  of  the  sphere  ;  find  the  volume  of  the  portion  of  the 
sphere  which  is  outside  of  the  cone.  4n'^^cos*  a 

3 

39.  Find  the  volume  produced  by  the  revolution  of  the  lemniscata 


about  a  perpendicular  to  the  initial  line.  Tr^a^  \/2 

8~ 


§  IX.]  EXAMPLES.  167 

40.  Find  the  volumes  generated  by  the  revolution  of  the  large  loop 
and  by  one  of  the  small  loops  of  the  curve 

r  ^=  a  cos  6  cos  29 

about  a  perpendicular  to  the  initial  line. 

-\ ,  and 


16  5  32         10  * 

41.  From  the  element 

r  dr  dB  dz 

derive  the  formulas  for  determining  the  volume  of  a  solid  of  revolution 
whose  axis  is  the  axis  of  z. 

V=-  2  7t    \rdrdz^ 

V=  Tt\(r^—rl)dz,  and  V  =  27t\(Zi  —  Zy)r  dr. 

Interpret  the  elements  in  these  integrals. 

42.  Find  the  volume  generated  by  the  revolution  of  the  curve 

in  which  a  >  b^  about  the  axis  oiy. 

Transform  to  polar  coordinates y  and  use  the  method  of  Art.  139. 

Ttb{2b''  +  3^')       na'  _^  b_ 

6  "^2V(«"-^')''^^     a' 

43.  Find  the  volume  generated  by  the  curve  given  in  the  preceding 
example,  when  revolving  about  the  axis  of  x. 

na  (2a'  +  ^b')  nb"  a  +  VC^"  -  b^) 


1 68  GEOMETRICAL  APPLICATIONS.  [Ex.  IX. 

44.  Find  the  volume  common  to  the  sphere  whose  radius  is  p  =  «, 
and  to  the  solid  formed  by  the  revolution  of  the  cardioid, 

r  =  «  (i  +  cos  0), 
about  the  initial  line. 

See  Art.  141.  — 7~  • 

45.  Find  the  whole  volume  enclosed  by  the  surface 

Transform  to  the  coordinates  p,  <^,  0,  and  show  that  the  solid  consists 

a' 
of  four  equal  detached  parts.  -7  • 


X. 

Rectification  of  Plane  Curves. 

(42.  A  curve  is  said  to  be  rectified  when  Its  length  is  deter- 
mined, the  unit  of  measure  to  which  it  is  referred  being  a 
right  line. 

It  is  shown  in  Diff.  Calc,  Art.  314  [Abridged  Ed.,  Art.  164], 
that,  if  s  denotes  the  length  of  the  arc  of  a  curve  given  in 
rectangular  coordinates,  we  shall  have 

ds  =  ^/{dx'  +  df\ 

If  the  abscissas  of  the  extremities  of  the  arc  are  known,  s  is 
found  by  substituting  for  dy  in  this  expression  its  value  in 
terms  of  x  and  dx^  and  integrating  the  result  between  the 
given  values  of  x  as  limits.  Thus,  to  express  the  arc  measured 
from  the  vertex  of  the  semi-cubical  parabola 


§  X.]  RECTIFICATION  OF  PLANE   CURVES.  1 69 

in  terms  of  the  abscissa  of  its  other  extremity,  we  derive,  from 
the  equation  of  the  curve, 

,        3  ^/x  dx 
dy=- — , 


whence  ds  —  — ^^^    ,         dx. 

2  Va 


Integrating, 


= \/(gx  +  4a)  dx 

2  f'a  Jo 


{gx  +  4ay  -  —  . 


2yVa^^  27 

(43.  When  x  and  y  are  given  in  terms  of  a  third  variable, 
ds  is  generally  expressed  in  terms  of  this  variable.  For  exam- 
ple, from  the  equations  of  the  four-cusped  hypocycloid, 

x  —  acos^ipy  y  =  asm^(/',     .     .     .     (i) 

we  derive 
dx  —  —  -i^a  cos^ ^  sin  ^  d^p,  and  dy  —  ^a  sin^  ^  cos  ^  dip\ 

whence  .      ds  =  2>^  sin  tp  cos  0  d^' (2) 

The  length  of  the  arc  between  the  point  {a,  o),  corresponding 
to  ip  —  o,  and  (o,  a)  corresponding  to  ^  =  |7r,  is  therefore 


I 


3^   •  <i 
--  sm 

2 


■  ■'■]:=? 


I/O  GEOMETRICAL  APPLICATIONS,  [Art.  1 44. 


Change  of  the  Sign  of  ds. 

I44-.  We  have  hitherto  assumed  ds  to  be  positive,  but  it  is 
to  be  remarked  that  an  expression  substituted  for  ds^  as  in  the 
illustration  given  in  the  preceding  article,  may  change  sign. 
Thus,  in  equation  (2),  ds,  which  is  so  written  as  to  be  positive 
while  ip  passes  from  o  to  \7t,  becomes  negative  while  ?/;  passes 
from  \n  to  n.  Thus  the  integral  gives  a  negative  result  for 
the  arc  between  the  points  (o,  a)  and  (—  ^,  o),  corresponding  to 
\n  and  n.  This  change  of  sign  in  ds  indicates  a  cusp  or  sta- 
iionary  point  of  the  curve ;  and  the  existence  of  such  points 
must  be  considered  before  we  can  properly  interpret  the  result- 
ing values  of  s.     For  instance,  if  in  this  example  we  integrate 

between  the  limits  o  and  — ,  we  get  the  result  ^  =  — ,  which  is 

4  4 

the  algebraic   sum,    but   the    numerical  difference   of   the   arcs 
between  the  points  corresponding  to  the  limits. 


Polar  Coordinates. 

146.  It   is  proved  in   Diff.  Calc,  Art.  317  [Abridged    Ed., 
Art.  167],  that  when  the  curve  is  given  in  polar  coordinates 

ds  =  Vidr"  +  r"  dffi). 

This   is   usually  expressed   in   terms  of  d.     For  example,   the 
equation  of  the  cardioid  is 

r  =  a  {i  —  cos  6)  =  2a  sin^ 1 0 ; 
whence  dr  =  2a  sin  ^0  cos  ^0  ^^, 

and  by  substitution 

ds  =  2a  sin  |-(9  d(^. 


§  X.]  RECTIFICATION  OF  CURVES,  I7I 

The  limits  for  the  whole  perimeter  of  the  curve  are  o  and  2;r, 
and  ds  remains  positive  for  the  whole  interval.     Therefore 


=  2a\    sin  —aO  =  —  Aa  cos  -       = 

Jo  2  ^  2J0 


8^. 


Rectification  of  Curves  of  Double  Curvature, 

14-6.  Let  G  denote  the  length  of  the  arc  of  a  curve  of  double 
curvature ;  that  is,  one  which  does  not  lie  in  a  plane,  and  sup- 
pose the  curve  to  be  referred  to  rectangular  coordinates  jir,  j/ 
and^.  If  at  any  point  of  the  curve  the  differentials  of  the 
coordinates  be  drawn  in  the  directions  of  their  respective  axes, 
a  rectangular  parallelopiped  will  be  formed,  whose  sides  are 
dx^  dy  and  dz^  and  whose  diagonal  is  da.     Hence 

da  =  V{d:^  +d/  ^  ds"). 

The  curve  is  determined  by  means  of  two  equations  connect- 
ing x^  y  and  ^,  one  of  which  usually  expresses  the  value  of  y  in 
terms  of  x,  and  the  other  that  of  z  in  terms  of  x.  We  can 
then  express  da  in  terms  of  x  and  dx. 

If  the  given  equations  contain  all  the  variables,  equations 
of  the  required  form  may  be  obtained  by  elimination. 

147.  An  equation  containing  the  two  variables  x  and  y 
only  is  evidently  the  equation  of  the  projection  upon  the  plane 
of  xy  of  a  curve  traced  upon  the  surface  determined  by  the 
other  equation.  Let  s  denote  the  length  of  this  projection  ; 
then,  since  ds^  —  dx^  -f-  df, 

da  =  Vids"  +  dz^, 

in  which  ds  may,  if  convenient,  be  expressed  in  polar  coordin- 
ates ;  thus, 

da  =  Vidr"  ^r^de^  +  dz% 


\^2  GEOMETRICAL  APPLICATIONS.  [Art.  148. 

!48,  As  an  illustration,  let  us  use  this  formula  to  deter- 
mine tiie  length  of  the  loxodromic  curve  from  the  equation  of 
the  sphere, 

x^  -V  f  ■\-  ^^a", (i) 

upon  which  it  is  traced,  and  its  projection  upon  the  plane  of 
the  equator,  of  which  the  equation  is 

or  in  polar  coordinates 

2a  — r  (e^^  4-  £-"^) (2) 

Equation  (i)  is  equivalent  to 

r^  +  ^  =  a^; 

and,  denoting   the  latitude  of  the  projected  point  by  #,  this 

gives 

js  =  asin  (l>,  r  —  a  cos  (j).     .     .     .     (3) 

In  order  to  express  dd  in  terms  of  ^,  we  substitute  the  value 
of  r  in  (2)  ;  whence 

£n6  _i_  g-n9  ~  2  SCC  (f>,       ......       {4) 

and  by  differentiation 

gne  __  ^-ne  ^  ?  sec  <^  tan  (f> -f^ (5) 

w  au 

Squaring  and  subtracting  equation  (5)  from  equation  (4), 
4  sec^  ^  r  2      .     2  JL  ^^1 

which  reduces  to 

M^=?^^^ (gj 


§  X.]  LENGTH  OF   THE  LOXODROMIC  CURVE.  1/3 


From  equations  (3)  and  (6) 

dr^  —  c?  sin^  ^  d<^^ 
d^  —  a^  cos^  (j)  d(p ; 

whence  substituting  in  the  value  of  da  (p.  171) 


da  =  aV[i  +-:^)d(!>. 
Integrating, 

ff  =  a  -^ '      d6  =  a -^ — , '  (6  —  a), 

n         ]a  n         ^'  " 

where  a  and  /?  denote   the    latitudes   of   the    extremities   of 
the  ar^. 

/  Examples  X. 

I.  Find  the  length  of  an  arc  measured  from  the  vertex  of  the 
catenary 


y-^U, 


-'  *  •'■"> 


and  show  that  the  area  between  the  coordinate  axes  and  any  arc  is 
proportional  to  the  arc. 

A  —  cs. 

^  2.  Find  the  length  of  an  arc  measured  from  the  vertex  of  the 

K)arabola 

y^  =  4ax. 

^(aar  f  x^)  -f  a  log -^ . 


J 


74  GEOMETRICAL  APPLICATIONS.  [Ex.  X. 


3.  Find  the  length  of  the  curve 

€^  +  I 


/^=- 


£-^  -  I  ' 

between  the  points  whose  abscissas  are  a  and  b. 

1      f  ^'^  —  I 

leg V  a  —  b, 

4.  Find  the  length,  measured  from  the  origin,  of  the  curve 


y=a\og 


«* 


,     a  ^-  X 

a  log X, 

°  a  —  X 


5.  Given  the  differential  equation  of  the  tractrix. 


and,  assuming  (o,  a)  to  be  a  point  of  the  curve,  find  the  value  of  s  as 
measured  from  this  point,  and  also  the  value  of  x  in  terms  oi  y  ;  that 
is,  find  the  rectangular  equation  of  the  curve. 

y 

s  —  a  log  —  . 


/ 


^  =  tf  log ^— -^-^  —  V(a  —  y). 


6.  Find  the  length  of  one  branch  of  the  cycloid 

X—  a(ip  —  sin ^),  y  =  a  {i  —  cos  tp). 

Sa. 

7.  When  the  cycloid  is  referred  to  its  vertex,  the  equations  being 

X  — a  {i  —  cosip)f  ^  =  « (^  +  sin  ?/?), 

prove  that  s  =  \/{Sax). 


§  X.]  EXAMPLES.  i;5 

8.  Find  the  length  from  the  point  (^,  o)  of  the  curve 

X  ^=  2a  cos  ip  —  a  cos  2tp, 

_y  =  2asmtp  —  a  sin  2tp. 

4a(tp  —  sin  tp). 

9.  Show  that  the  curve, 

X  =  ^a  cos  'Z'  —  2a  cos* ^,  y=  2a  sin'  ^^, 

has  cusps  at  the  points  given  by  i/^  =  o  and  tp  =  tt  ;  and  find  the 
whole  length  of  the  curve.  12a. 


10.  Find  the  length  of  a  quadrant  of  the  curve 


See  Ftg.  6,  Art.  107.  -—  -  , 

a  ^r  o 

II.  Show  that  the  curve 

x=^  2a  cos*  ^  (3  —  2  cos^  ^),  j;  =  4^  sin  ^  cos^  ^ 

has  three  cusps,  and  that  the  length  of  each  branch  is  —  . 

/ 

J    12.  Find  the  length  of  the  arc  between  the  points  at  which   the 
curve 

A' =  ^cos'^  ^cos2^,  _;' =«sin^^sin  26^ 

2-i/2 

cuts  the  axes.  .  a. 


' — r~  ^^^ 


.;^ 


GEOMETRICAL   APPLICATIONS.  [Ex.  X. 


N     13.  3how  that  the  curve 
\A*^  xJ^  ^  ^  —  (^  cos  ^  (i  +  sin"  ^), 


^X 


y  =  a^xntp  cos^*  ^ 


is  symmetrical  to  the  axes,  and  find  the  length  of  the  arcs  between 

the  cusps.  /   ,  .  I 

^  •       a  [  \/2  —  sm-^  — 

\  V3 

a[  4/2  '+  cos"^  — -  J  . 
\  4/3/ 

\       14.  Find  the  length  of  one  branch  of  the  epicycloid 

/         7v  /        7        a  -\r  b  . 

X  =  [a  -i-  P)  cos  ip  —  o  cos  — 7 —  ^, 


^  =  (^  +  ^)  sin  ^  —  ^  sin  — - —  ^. 


U  {a  +  b) 
a 


15.  Show  that  the  curve 


X  =  ga  sin  tp  —  4a  sin'  ^, 
y  =  —  3<a!  cos  rp  +  4a  cos'  ip 

is  symmetrical  to  the  axes,  and  has  double  points  and  cusps  :  find  the 
lengths  of  the  arcs,  (a)  between  the  double  points,  (/?)  between  a 
double  point  and  a  cusp,  and  (y)  the  arc  connecting  two  cusps,  and  not 
passing  through  the  double  points. 

\     16.  Find  the  whole  length  of  the  curve 

X  =  $a  sin  if^  —  a  sin'  ^, 

y=  a  cos'  t/j.  ^Tta. 


EXAMPLES.  177 


17.   Find  the  length,  measured  from  the  pole,  of   any  arc  of  the 
equiangular  spiral 


in  which  n  =  cot  a.  r  sec  a 


r  —  aB^  , 


18.  Prove  by  integration  that  the  arc  subtending  the  angle  9  at  the 
circumference  in  a  circle  whose  radius  is  «,  is  zafj. 

19.  Find  the  length,  measured  from  the  origin,  of  the  curve  defined 
by  the  equations 


2a'  6a" 


6d' 


20.  Find  the  length,  measured  from  the  origin,  of  the  intersection  of 
the  surfaces 

J'  =  4«  sin  Xy  z  =^  271  {2x  -\-  sin  2x). 

{4n^  +  i):^:^  +  2^'*  sin  2X. 

21.  Find  the  length,  measured  from  the  origin,  of  the  intersection  of 
the  cylindrical  surfaces 

[y  —  xY  =  4aXf  ga  (z  —  xY  =  4^•^ 


2x2 


+  2  V{ax)+x. 


22.  If  upon  the  hyperboHc  cylinder 

c'       b'       '' 
a  curve  whose  projection  upon  the  plane  of  xy  is  the  catenary 

X  X 

be  traced,  prove  that  any  arc  of  the  curve  bears  to  the  corresponding 
arc  of  its  projection  the  constant  ratio  V(b^  +  /■*)  :  c. 


78 


GEOMETRICAL   APPLICATIONS. 


[Art.  149 


XI. 


Surfaces  of  Solids  of  Revolution, 

149-  The  surface  of  a  solid  of  revolution  may  be  generated 
by  the  circumference  of  the  circular  section  made  by  a  plane 

perpendicular  to  !he  axis  of  revolu- 
tion. Thus  in  Fig.  17,  the  surface 
produced  by  the  revolution  of  the 
curve  AB  about  the  axis  of  x  is  re- 
garded as  generated  by  the  circum- 
ference PQ.  The  radius  of  this  cir- 
cumference is  J,  and  its  plane  has  a 
motion  whose  differential  is  dx,  but 
every  point  in  the  circumference  itself 
has  a  motion  whose  differential  is  ds,s 
denoting  an  arc  of  the  curve  AB. 
Hence,  denoting  the  required  surface  by  5,  we  have 

dS=  27ty  ds=  27ty  i/(dx^  +  <^). 

The  value  of  dS  must  of  course  be  expressed  in  terms  of  a  single 
variable  before  integration. 

150.  For  example,  let  us  determine  the  area  of  the  zone  of 
spherical  surface  included  between  any  two  parallel  planes. 
The  radius  of  the  sphere  being  a,  the  equation  of  the  revolv- 
ing curve  is 

X^  +  f^=  a^; 

whence  y  =    V{d^  —  -^'^), 

_  xdx 

adx 


and 


dS  =  27r^  dx ; 


§  XL]  SURFACES   OF  SOLIDS    OF  REVOLUTION.  1 79 


therefore 


S  =  2na\dx  =^  27ta  (xi  —  x^ 


Since  x<i,  —  Xi  is  the  distance  between  the  parallel  planes, 
the  area  of  a  zone  is  the  product  of  its  altitude  by  27ta,  the 
circumference  of  a  great  circle,  and  the  area  of  the  whole  sur- 
face of  the  sphere  is  47ra^. 

151.  When  the  curve  is  given  in  polar  coordinates,  it  is  con- 
venient to  transform  the  expression  for  vS"  to  polar  coordinates. 
Thus,  if  the  curve  revolves  about  the  initial  line, 


S  =  27t{fc/s=  2;r  r  sin  ^  V{c^r^  +  ^ 


For  example,  if  the  curve  is  the  cardioid 
we  find,  as  in  Art.  145, 


r  =:2^sin^— ^ , 


Hence 


ds  =  2a  sin  —  B  dd. 
2 


("I  I 

sin*-  ^cos-  0 
2  2 


dd 


^_ — _  sm^-  f      —  ^ 


Areas  of  Surfaces  in   General, 

162.  Let  a  surface  be  referred  to  rectangular  coordinates  x^ 
y  and  z ;  the  projection  of  a  given  portion  of  the  surface  upon 
the  plane  of  xy  is  a  plane  area  determined  by  a  given  relation 
between  x  and  y.  We  may  take  as  the  elements  of  the  surface 
the   portions   which   are   projected    upon    the    corresponding 


8o 


GEOMETRICAL   APPLICATIONS. 


[Art.  152. 


elements  of  area  in  the  plane  of  xy.  If  at  a  point  within  the 
element  of  surface,  which  is  projected  upon  a  given  element 
i\x i\y^  a  tangent  plane  be  passed,  and  if  y  denote  the  inclina- 
tion of  this  plane  to  the  plane  of  xy,the  area  of  the  correspond- 
ing element  in  the  tangent  plane  is 

sec  y  A  X  Ajj/. 

The  surface  is  evidently  the  limit  of  the  sum  of  the  elements 
in  the  tangent  planes  when  ax  and  Aj/  are  indefinitely  dimin- 
ished. Now  sec  ;^  is  a  function  of  the  coordinates  of  the  point 
of  contact  of  the  tangent  plane ;  and  since  these  coordinates 
are  values  of  x  and  y  which  lie  respectively  between  x  and 
X  +  AX  and  between  j^  and y  +  A/,  the  theorem  proved  in  Art. 
99  shows  that  this  limit  is 


5  =      sec  ;/  dx  dy. 


153.  The  value  of  sec  y   may  be   derived  by  the  following 

method.     Through  the  point  P  of 
the   surface  let   planes    be    passed 
parallel  to  the    coordinate    planes, 
and  let  PD,  and  PE,  Fig.  17,  be  the 
intersections  of  the  tangent   plane 
with    the    planes   parallel    to    the 
planes  of  xz  andj^^.     Then  PD  and 
PE  are  tangents  at  P  to  the  sec- 
tions of  the  surface  made  by  these 
planes.     The    equations    of    these 
sections  are  found  by  regarding  y 
and  X  in  turn  as  constants  in  the  equation  of  the  surface  ;  there- 
fore denoting  the  inclinations  of  these  tangent  lines  to  the  plane 
of  xy  by  ^  and  ^,  we  have 


Fig.  18. 


tan  0  = 


dx' 


and 


tan '/'  — 


dz 
dy' 


§  XL]  AREAS  OF  SURFACES  IN  GENERAL.  l8l 

/J'^  {123 

in  which  — -  and-r  are  partial  derivatives  derived  from  the  equa- 
dx         ay 

tion  of  the  surface. 

If  the  planes  be  intersected  by  a  spherical  surface  whose 
centre  is  P,  ADE  is  a  spherical  triangle  right  angled  at  A, 
whose  sides  are  the  complements  of  ^  and  ^.  Moreover,  if  a 
plane  perpendicular  to  the  tangent  plane  PED  be  passed 
through  AP^  the  angle  FPG  will  be  y,  and  the  perpendicular 
from  the  right  angle  to  the  base  of  the  triangle  the  comple- 
ment of  y. 

Denoting  the  angle  EAF  by  (9,  the  formulas  for  solving 
spherical  right  triangles  give 

^      tan  ^  J  .     n      tan  i> 

cos  Q  = ,  and  sm  a  = . 

tan  y  tdiVi  y 

Squaring  and  adding, 

_  tan^  ip  +  tan^  (f> 
~~  tan^  y         ' 

or  tan^  y  =  tan*^  tf)  +  tan^  (p ; 
whence  sec'  y=i+  (J^)'  +  (|)'. 

Substituting  in  the  formula  derived  in  Art.  (152),  we  have 

154.  It  is  sometimes  more  convenient  to  employ  the  polar 


1 82  GEOMETRICAL  APPLICATIONS.  [Art.  1 54. 

element  of  the  projected  area.     Thus  the  formula  becomes 

5  =      sec  yrdrdOy 

where  sec  y  has  the  same  meaning  as  before. 

For  example,  let  it  be  required  to  find  the  area  of  the  sur- 
face of  a  hemisphere  intercepted  by  a  right  cylinder  having  a 
radius  of  the  hemisphere  for  one  of  its  diameters.  From  the 
equation  of  the  sphere, 

x^+f  +  ^=d\ (i) 

we  derive 

dz  _       X  dz  _      y 

dx  z^  dy  z  * 


whence 


--=^[■^©•-(1) 


therefore         *  5        '  ' 


-«jf 


the  integration  extending  over  the  area  of  the  circle 

r  =  a  cos  6 (2) 

Since  equation  (i)  is  equivalent  to 

^  +  r2  =  ^, 


§  XI.]  AREAS  OF  SURFACES  IN  GENERAL.  1 83 

From  (2)  the  limits  for  ;■  are  ri  =  o,  and  r^  —  a  cos  ^, 
hence 


5r=^|(i  -sin6')^/9, 


in  which  a  sin  Q  is  put  for  \.\\^ positive  quantity  s/(c?  —  r}).  The 
limits  for  Q  are  —\'n:  and  \7i^  but  since  sin  /9  is  in  this  case  to 
be  regarded  as  invariable  in  sign,  we  must  write 


5  ■=  2^2  [  '(I  _  sin  B)  dO  =  na^  -  2a\ 

Jo 


If  another  cylinder  be  constructed,  having  the  opposite  radius 
of  the  hemisphere  for  diameter,  the  surface  removed  is 
27ta^  —  4a^,  and  the  surface  which  remains  is  4^?^  a  quantity 
commensurable  with  the  square  of  the  radius.  This  problem 
was  proposed  in  1692,  in  the  form  of  an  enigma,  by  Vivian i,  a 
Florentine  mathematician. 


Examples  XL 

I.  Find  the  surface  of  the  paraboloid  whose  altitude  is  a-:,  and  the 
radius  of  whose  base  is  d. 


2.  Prove  that  the  surface  generated  by  the  arc  of  the  catenary  given 
in  Ex.  X.,  I,  revolving  about  the  axis  of  x,  is  equal  to 

7t{cx  +  sy). 

3.  Find  the  whole  surface  of  the  oblate  spheroid  produced  by  the 


84  GEOMETRICAL  APPLICATIONS.  [Ex.  XI. 


revolution  of  an  ellipse  about  its  minor  axis,  a  denoting  the  major, 
b.  the  minor  semi-axis,  and  e  the  excentricity,  — ^^ . 

2    .        b\      1  +e 
27ta    -i-  TV  -  log -. 


4.  Find  the  whole  surface  of  the  prolate  spheroid  produced  by  the 
revolution  of  the  ellipse  about  its  major  axis,  using  the  same  notation 
as  in  Ex.  3. 

,2   ,           -sin"^^ 
2  7to   -}-  2  7rab . 


5.  Find  the  surface  generated  by  the  cycloid 

X  z=  a  {(p  —  sin  //?),         y  =  a  {i  —  cos  //') 


revolving  about  its  base.  —  7r«^ 

3 


6.  Find  the  surface  generated  when  the  cycloid  revolves  about  the 
tangent  at  its  vertex. 


7.  Find  the  surface  generated  when  the  cycloid  revolves  about  its 
axis. 


8.  Find  the  surface  generated  by  the  revolution  of  one  branch  of 
the  tractrix  (see  Ex.  X.,  5)  about  its  asymptote. 


§  XL]  EXAMPLES.  185 

9.  Find  the  surface  generated  by  the  revolution  about  the  axis  of 
X  of  the  portion  of  the  curve 

y  =  f  ^ 

which  is  on  the  left  of  the  axis  of  y. 

7r[  |/2  +  log  (i  +   V2)]. 

10.  Find  the  surface  generated  by  the  revolution  about  the  axis  of 
X  of  the  arc  between  the  points  for  which  x  =  a  and  :xr  =  ^  in  the 
hyperbola 

xy  =  k^. 


II.  Show  that  the  surface  of  a  cylinder  whose  generating  lines  are 
parallel  to  the  axis  of  z  is  represented  by  the  integral 


5 


=  \z  t/Sj 


where  s  denotes  the  arc  of  the  base  in  the  plane  of  xy.  Hence, 
deduce  the  surface  cut  from  a  right  circular  cylinder  whose  radius  is 
a,  by  a  plane  passing  through  the  centre  and  making  the  angle  ^  with 
the  plane  of  the  base.  za^  tan  a. 

12.  Find  the  surface  of  that  portion  of  the  cylinder  in  the  problem 
solved  in  Art.  154,  which  is  within  the  hemisphere.  20^. 

13.  Find  the  surface  of  a  circular  spindle,  a  being  the  radius  and 
2c  the  chord. 


47ra 


c-   t/(a'^-^=^jsin-- 


1 86  GEOMETRICAL  APPLICATIONS.  [Art.  1 5 5, 


XII. 

The  Area  generated  by  a  Straight  Line  moving  in  any 
Manner  in  a  Plane, 

155.  If  a  straight  line  of  indefinite  length  moves  in  any  man- 
ner whatever  in  a  plane,  there  is  at  each  instant  a  point  of  the 
line  about  which  it  may  be  regarded  as  rotating.  This  point  we 
shall  call  the  centre  of  rotation  for  the  instant.  The  rate  of 
motion  of  every  point  of  the  line  in  a  direction  perpendic- 
ular to  the  line  itself  is  at  the  instant  the  same  as  it  would 
be  if  the  line  were  rotating  at  the  same  angular  rate  about  this 
point  as  a  fixed  centre."^  Hence  it  follows  that  the  area 
generated  by  a  definite  portion  of  the  line  has  at  the  instant 
the  same  rate  as  if  the  line  were  rotating  about  a  fixed  instead 
of  a  variable  centre. 

(56.  Suppose  at  first  that  the  centre  of  rotation  is  on  the 
generating  line  produced,  p^  and  p^  denoting  the  distances  from 
the  centre  of  the  extremities  of  the  generating  line,  and  let  <i> 
denote  its  inclination  to  a  fixed  line.  By  substitution  in  the 
general  formula  derived  in  Art.  no,  we  have 

dA  =Yp}-p^)cU. 

♦Compare  Difif.  Calc,  Art.  332  [Abridged  Ed.,  Art.  176J,  where  the  moving 
line  is  the  normal  to  a  given  curve,  and  the  centre  of  rotation  is  the  centre  of  cur- 
vature of  the  given  curve.  If  the  line  is  moving  without  change  of  direction,  the 
centre  is  of  course  at  an  infinite  distance. 

When  the  line  is  regarded  as  forming  a  part  of  a  rigidly  connected  system  in 
motion,  its  centre  of  rotation  is  the  foot  of  a  perpendicular  dropped  upon  it  from 
the  instantaneous  centre  of  the  motion  of  the  system.  Thus,  if  the  tangent  and 
normal  in  the  illustration  cited  are  rigidly  connected,  the  centre  of  curvature,  C,  is 
the  instantaneous  centre  of  the  motion  of  the  system,  and  the  point  of  contact,  P, 
is  the  centre  of  rotation  for  the  tangent  line. 


§  XII.]  AREAS  GENERATED  BY  MOVING  LINES.  1 8/ 


Applications, 

157.  The  area  between  a  curve  and  its  evolute  may  be 
generated  by  the  radius  of  curvature  p,  whose  inclination  to 
the  axis  of  ;ir  is  ^  +  ^n^  in  which  (j)  denotes  the  inclination 
of  the  tangent  line.  Since  the  centre  of  rotation  is  one 
extremity  of  the  generating  linep,  the  differential  of  this  area 
is  found  by  substituting  in  the  general  expression  Pi  =  o  and 
P2  =  o.     Hence  when  p  is  expressed  in  terms  of  ^, 


^  =  Mp2^0 


expresses  the  area  between  an  arc  of  a  given  curve,  its  evolute, 
and  the  radii  of  curvature  of  its  extremities,  the  limits  being 
the  values  of  (t>  at  the  ends  of  the  given  arc. 

158.  For  example,  in  the  case  of  the  cardioid 

r  —  a{i  —  cos  6), 

it  is  readily  shown,  from  the  results  obtained  in  Art.  145,  that 
the  angle  between  the  tangent  and  the  radius  vector  is  ^6\  and 
therefore  ^  =  |(9,  and 

ds       Aa   .    6 

To  obtain  the  whole  area  between  the  curve  and  its  evolute, 
the  limits  for  B  are  o  and  27t ;  hence  the  limits  for  ^  are  o 
and  37r.     Therefore 


A 


'%^d^=^J^r^r.^td^  =  ^^, 


159.  As    another   application    of    the    general    formula    of 
Art.  156,  let   one  end   of  a  line   of  fixed  length  a  be  moved 


1 88  GEOMETRICAL  APPLICATIONS,  [Art.  1 59. 

along  a  given  line  in  a  horizontal  plane,  while  a  weight  at- 
tached to  the  other  extremity  is  drawn  over  the  plane  by  the 
line,  and  is  therefore  always  moving  in  the  direction  of  the 
line  itself.  The  line  of  fixed  length  in  this  case  turns  about 
the  weight  as  a  moving  centre  of  rotation.  Hence  the  area 
generated  while  the  line  turns  through  a  given  angle  is  the 
same  as  that  of  the  corresponding  sector  of  a  circle  whose 
radius  is  a. 

The  curve  described  by  the  weight  is  called  a  tractrix^  and 
the  line  along  which  the  other  extremity  is  moved  is  the  direc- 
trix. When  the  axis  of  x  is  the  directrix,  and  the  weight 
starts  from  the  point  (o,  a),  the  common  tractrix  is  described ; 
hence  the  area  between  this  curve  and  the  axis  is  ^na^. 

160.  Again,  in  the  generation  of  the  cycloid,  Diff.  Calc, 
Art.  288  [Abridged  Ed.,  Art.  156],  the  variable  chord  RP  may 
be  regarded  as  generating  the  area.  The  point  R  has  a  motion 
in  the  direction  of  the  tangent  RX\  the  point  P  partakes  of 
this  motion,  which  is  the  motion  of  the  centre  Cj  and  also  has 
an  equal  motion,  due  to  the  rotation  of  the  circle  in  the  direc- 
tion of  the  tangent  to  the  circle  at  P.  Since  the  tangents 
at  P  and  R  are  equally  inclined  to  PRy  the  motion  of  P  in  a 
direction  perpendicular  to  PR  is  double  the  component,  in  this 
direction,  of  the  motion  of  R,  Therefore  the  centre  of  rota- 
tion of  PR  is  beyond  i?  at  a  distance  from  it  equal  to  PR, 
Hence,  denoting  PRO  by  ^, 

Pi  —  PR  —  2a  sin  <^,  pg  =  '2-PR  =  4^  sin  ^. 

Substituting  in  the  formula  of  Art.  156,  we  have  for  the  area 
of  the  cycloid,  since  PRO  varies  from  o  to  n^ 


A  =  6^^      sin^  ^  d<l>  —  -^nc?. 


§  XIL] 


SIGN  OF   THE   GENERATED  AREA. 


189 


Sign  of  the  Generated  Area. 

(61.  Let  AB  be  the  generating  line,  and  ^  the  centre  ol 
rotation.     The  expression, 


dA  =  i  {Pi  -  P?)  d(l>, 


(0 


Fig.  ig. 


for  the  differential  of  the  area,  was  obtained  upon  the  supposi- 
tion that  A  and  B  were  on  the  same  side  of  C.     Then  suppos- 
ing P2  >  Pi,  and  that  the  line  rotates   in  the  positive  direction^ 
as  in  figure  19,  the  differential  of  the  area  is 
positive;  and  we  notice  that  every  point  in  the 
area   generated   is  swept    over   by  the  line 
AB,    the  left  hand  side  as  we  face   in  the 
direction  A  B  preceding. 

162.  We  shall  now  show  that  in   every 
case,    the    formula   requires    that    an    area 
swept  over  with    the  left   side  preceding,  shall  be  considered 
as  positively  generated,  and    one  swept  over  in  the  opposite 
direction  as  negatively  generated. 

In  the  first  place,  if  C  is  between  A  and 
B  so  that  p,  is  negative,  as  in  figure  20,  p^ 
is  still  positive,  and  formula  (i)  still  gives 
the  difference  between  the  areas  generated 
hy  AB  and  AC.  Hence  the  latter  area, 
which  is  now  generated  by  a  part  of  the 
line  AB,  must  be  regarded  as  generated 
negatively,  but  the  right  hand  side  as  we 
face  in  the  direction  AB  of  this  portion  of  the  line  is  now 
preceding,  which  agrees  with  the  rule  given  in  Art.  161. 

Again,  if  C  is  beyond  B,  the  formula  gives  the  difference 
of  the  generated  areas ;  but  since  pi  is  numerically  greater 
than  p2^,  in  this  case,  dA  is  negative,  and  the  area  generated  by 
AB  is  the  difference  of  the  areas,  and  is  negative  by  the  rule. 


Fig.  20. 


190 


GEOMETRICAL   APPLICATIONS. 


[Art.  162. 


Finally,  if  the  direction  of  rotation  be  reversed,  d(l>  and 
therefore  dA  change  sign,  but  the  opposite  side  of  each  por- 
tion of  the  line  becomes  in  this  case  the  preceding  side. 

163.  We  may  now  put  the  expression  for  the  area  in  another 
form.     For 


dA=~{pl-ri)d^ 


(Ps-pO^^V^; 


whatever  be  the  signs  of  Pg  and  Pi,  the  first  factor  is  the  length 
of  AB,  which  we  shall  denote  by  /,  and  the  second  factor  is 
the  distance  of  the  middle  point  of  AB  from  the  centre  of 
rotation,  which  we  shall  denote  by  p^„.     Hence,  putting 


P2  -  Pi  =  /, 
we  have 


and 


Ipm  d(l>. 


P2+   Px__ 


(2) 


Since  p^  d(j>  is  the  differential  of  the  motion  of  the  middle  point 
in  a  direction  perpendicular  to  AB,  this  expression  shows  that 
the  differential  of  the  area  is  the  product  of  this  differential  by 
the  length  of  the  generating  line. 


Areas  generated   by  Lines  whose   Extremities   describe 
Closed  Circuits. 


i-^ 


I64-.  Let  us  now  suppose  the  generating  line  AB  to  move 
from  a  given  position,  and  to  return  to  the 
same  position,  each  of  the  extremities  A  and 
B  describing  a  closed  curve  in  the  positive 
direction,  as  indicated  by  the  arrows  in  figure 
21.  It  is  readily  seen  that  every  point  which 
is  in  the  area  described  by  B,  and  not  in  that 
described  by  A^  will  be  swept  over  at  least 
once  by  the   line  AB^  the  left  side  preceding, 

Fig.  21.  and  if  passed  over  more  than  once,  there  will  be 


§  XII.]        AREAS  GENERATED  BY  MOVING  LINES,  I9I 

an  excess  of  one  passage,  the  left  side  preceding.  Therefore 
the  area  within  the  curve  described  by  i?,  and  not  within  that 
described  hy  A,  will  be  generated  positively.  In  like  manner 
the  area  within  the  curve  described  by  A,  and  not  within  that 
described  by  B,  will  be  generated  negatively.  Furthermore,  all 
points  within  both  or  neither  of  these  curves  are  passed  over, 
if  at  all,  an  equal  number  of  times  in  each  direction,  so  that  the 
area  common  to  the  two  curves  and  exterior  to  both  disap- 
pears from  the  expression  for  the  area  generated  by  AB. 

Hence  it  follows  that,  regarding  a  closed  area  whose  perimeter 
is  described  in  the  positive  direction  as  positive^  the  area  generated 
by  a  line  returning  to  its  original  position  is  the  difference  of  the 
areas  described  by  its  extremities.  This  theorem  is  evidently 
true  generally,  if  areas  described  in  the  opposite  direction  are 
regarded  as  negative. 


Amslers  Planimeter, 

165.  The  theorem  established  in  the  preceding  article  may 
be  used  to  demonstrate  the  correctness  of  the  method  by 
which  an  area  is  measured  by  means  of  the  Polar  Planimeter, 
invented  by  Professor  Amsler,  of  Schaffhausen. 

This  instrument  consists  of  two  bars,  OA  and  AB,  Fig.  22, 
jointed  together  at  A,  The  rod  OA  turns  on 
a  fixed  pivot  at  (9,  while  a  tracer  at  B  is  carried 
in  the  positive  direction  completely  around 
the  perimeter  of  the  area  to  be  measured.  At 
some  point  C  of  the  bar  AB  a  small  wheel  is 
fixed,  having  its  axis  parallel  to  AB,  and  its 
circumference  resting  upon  the  paper.  When 
^is  moved,  this  wheel  has  a  sliding  and  a  roll- 
ing motion  ;  the  latter  motion  is  recorded  by 
an  attachment  by  means  of  which  the  number  Fig.  22. 

of  turns  and  parts  of  a  turn  of  the  wheel  are  registered. 


192  GEOMETRICAL   APPLICATIONS.  [Art.   166. 

166.  Let  J/ be  the  middle  point  of  AB,  and  let 
OA  =a,  AB  =  b,  MC^c. 

Since  b  is  constant,  the  area  described  hy  AB  is  by  equation  (2), 
Art.  163, 


\9md(l> (l) 


Kx^2.AB  —  b 


Denoting  the  linear  distance  registered  on  the  circumference 
of  the  wheel  by  s^  ds  is  the  differential  of  the  motion  of  the 
point  C,  in  a  direction  perpendicular  to  AB^  and  since  the  dis- 
tance of  this  point  from  the  centre  of  rotation  is  Pni  +  ^, 

ds  =  {p„t  +  c)  d(f) : 
substituting  in  (i)  the  value    of  pmd(l>y 

ArGSiAB  =  b{ds-bAd^ (2) 

167.  Two  cases  arise  in  the  use  of  the  instrument.  When, 
as  represented  in  Fig.  22,  O  is  outside  the  area  to  be  meas- 
ured, the  point  A  describes  no  area,  and  by  the  theorem  of 
Art.    164,   equation   (2)   represents  simply   the  area   described 

by  B.     In  this  case  ^  returns  to  its  original  value,   hence    d(f> 

vanishes,  and  denoting  the  area  to  be  measured  by^,  equation 
(2)  becomes 

A  =  6s (3) 

In  the  second  case,  when  O  is  within  the  curve  traced  by  By 
the  point  A  describes  a  circle  whose  area  is  Ttd^j  and  the  limit- 


§  XII.]  AMSLER'S  PLANIMETER.  193 

ing  values  of  <i>  differ  by  a  complete  revolution.  Hence  in  this 
case  equation  (2)  becomes 

A  —  ncP-  —  bs  —  2  Ttbcy 

or  A=  bs  ^-  7c{c?  —  2bc)!^ (4) 

In  another  form  of  the  planimeter  the  point  A  moves  in  a 
straight  line,  and  the  same  demonstration  shows  that  the  area 
is  always  equal  to  bs. 

Examples   XII. 

I.  The  involute  of  a  circle  whose  radius  is  a  is  drawn,  and  a  tangent 
is  drawn  at  the  opposite  end  of  the  diameter  which  passes  through  the 
cusp  ;  find  the  area  between  the  tangent  and  the  involute. 

a'n  (3  +  n'') 


2.  Two  radii  vectoresof  a  closed  oval  are  drawn  from  a  fixed  point 
within,  one  of  which  is  parallel  to  the  tangent  at  the  extremity  of  the 
other ;  if  the  parallelogram  be  completed,  the  area  of  the  locus  of  its 
vertex  is  double  the  area  of  the  given  oval. 

3.  Show  that  the  area  of  the  locus  of  the  middle  point  of  the  chord 
joining  the  extremities  of  the  radii  vectores  in  Ex.  2,  is  one  half  the 
area  of  the  given  oval. 


*  The  planimeter  is  usually  so  constructed  that  the  positive  direction  of  rotation 
is  with  the  hands  of  a  watch.  The  bar  b  is  adjustable,  but  the  distance  y^  C  is  fixed 
so  that  c  varies  with  b.  Denoting  AChy  q,  we  have  c  =  q  —  \b,  and  the  constant 
to  be  added  becomes  C  =^  it  {a^  —  ibq  -\-  b'^)  in  which  a  and^  are  fixed  and  b  adjusta- 
ble.     In  some  instruments  q  is  negative. 

It  is  to  be  noticed  that  in  the  second  case  s  may  be  negative  ;  the  area  is  then 
the  numerical  difiference  between  the  constant  and  bs. 


194  GEOMETRICAL  APPLICATIONS.  [Ex.  XII. 

4.  Prove  that  the  difference  of  the  perimeters  of  two  parallel  ovals, 
whose  distance  is  3,  is  2  nb,  and  that  the  difference  of  their  areas  is  the 
product  of  b  and  the  half  sum  of  their  perimeters. 

5.  A  lima9on  is  formed  by  taking  a  fixed  distance  be  on  the  radius 
vector  from  a  point  on  the  circumference  of  a  circle  whose  radius  is  a  ; 
show  that  the  area  generated  by  b  when  b'>  2«  is  the  area  of  the  lima- 
9on  diminished  by  twice  the  area  of  the  circle,  and  thence  determine 
the  area  of  the  lima9on. 

7r(2^^  +  b''), 

6.  Verify  equation  (4),  Art.  167,  when  the  tracer  describes  the 
circle  whose  radius  \^  a  -\-  b. 

7.  Verify  the  value  of  the  constant  in  equation  (4),  Art.  167,  by 
determining  the  circle  which  may  be  described  by  the  tracer  without 
motion  of  the  wheel. 

8.  If,  in  the  motion  of  a  crank  and  connecting  rod  (the  line  of  motion 
of  the  piston  passing  through  the  centre  of  the  crank),  Amsler's  record- 
ing wheel  be  attached  to  the  connecting  rod  at  the  piston  end,  deter- 
mine s  geometrically,  and  verify  by  means  of  the  area  described  by  the 
other  end  of  the  rod. 

9.  The  length  of  the  crank  in  Ex,  8  being  a^  and  that  of  the  con- 
necting rod  b,  find  the  area  of  the  locus  of  a  point  on  the  connecting 
rod  at  a  distance  c  from  the  piston  end. 


10.  If  a  line  AB  of  fixed  length  move  in  a  plane,  returning  to  its 
original  position  without  making  a  complete  revolution^  denoting  the  areas 
of  the  curves  described  by  its  extremities  by  {A)  and  (^),  determine 
the  area  of  the  curve  described  by  a  point  cutting  AB  in  the  ratio 
m  :  n. 

n{A)  4-  m(B) 
m  +  n 


§  XII.] 


EXAMPLES. 


195 


II.  If  the  line  in  Ex.  10  return  to  its  original  position  after  making  a 
complete  revolution^  prove  Holditch's  Theorem  j  namely,  that  the  area  of 
the  curve  described  by  a  point  at  the  distance  c  and  c  from  A  and  B  is 


c'{A)  +  c(B) 


c  ■{-  c' 


ncc 


12.  Show  by  means  of  Ex.  11  that,  if  a  chord  of  fixed  length  move 
around  an  oval,  and  a  curve  be  described  by  a  point  at  the  distances 
c  and  c  from  its  ends,  the  area  between  the  curves  will  be  ttcc  . 


XIII. 
Approximate  Expressions  for  Areas  and  Volumes. 

168.  When  the  equation  of  a  curve  is  unknown,  the  area 
between  the  curve,  the  axis  of  x,  and 
two  ordinates  may  be  approximately  ex- 
pressed in  terms  of  the  base  and  a  lim- 
ited number  of  ordinates,  which  are  sup- 
posed to  have  been  measured. 

Let  ABCDE  be  the  area  to  be  de- 
termined ;  denote  the  length  of  the  base 
by  2h ;  and  let  the  ordinates  at  the  ex- 
tremities and  middle  point  of  the  base 
be  measured  and  denoted  by  y^.y^,  and  jj/g.  Taking  the  base  for 
the  axis  of  x,  and  the  middle  point  as  origin,  let  it  be  assumed 
that  the  curve  has  an  equation  of  the  form 

y^  A  ^  Ex  ^  C^-V  D:^ ; (i) 


Fig.  23. 


then  the  area  required  is 

^       V'       ^        ^        Bj^      C^      DxT 

A  —        ydx-Ax^-  —  H +  

J-/  2  3  4   _ 

in  which  which  A  and  C  are  unknown 


''   ^-{(yA^2Ch%    .  (2) 

-h        3 


ig6 


GEOMETRICAL  APPLICATIONS. 


[Art.  i68. 


In   order  to  express   the    area  in   terms    of  the   measured 
ordinates,  we  have  from  equation  (i), 

whence  we  derive 


and  substituting  in  (2), 


A 


{yi  +  4j2  +  J/3). 


It  will  be  noticed  that  this  formula  gives  a  perfectly  ac- 
curate result  when  the  curve  is  really  a  parabolic  curve  of  the 
third  or  a  lower  degree. 

169.  If  the  base  be  divided  into  three  equal  intervals,  each 
denoted  by  k,  and  the  ordinates  at  the  extremities  and  at  the 
points  of  division  measured,  we  have,  by  assuming  the  same 
equation, 

A=\^\^jdx=^-^{AA  +  iOf) (I) 


From  the  equation  of  the  curve, 

y,=A 


Fig.  24. 


3^^       gCh^       27Dh^ 


,       Bh      Ch^      Dh^ 
y,^A-^^----^- 

,      Bh      Ch^      Dk^ 


^4 


^  4.  3^  4-  9^'  +  ?Z^'  ; 


§  XIII.]  SIMPSON'S  RULES.  1 97 


+  9f^. 


whence  ji  +  }\  =  2  A 


From  these  equations  we  obtain 


i6 


and  a^=^^--'^^-^A^V 

4 

Substituting  in  equation  (i), 


A  =^iyi+  3J2+  3/8 +J4). 


Simpson  s  Rules. 

!70.  The  formulas  derived  in  Articles  168  and  169,  although 
they  were  first  given  by  Cotes  and  Newton,  are  usually  known 
as  Simpsons  Rules,  the  following  extensions  of  the  formulas 
having  been  published  in  1743,  in  his  Mathematical  Disserta- 
tions. 

If  the  whole  base  be  divided  into  an  even  number  n  of 
parts,  each  equal  to  h,  and  the  ordinates  at  the  points  of  divis- 
ion be  numbered  in  order  from  end  to  end,  then  by  applying 
the  first  formula  to  the  areas  between  the  alternate  ordinates, 
we  have 

That  is  to  say,  the  area  is  equal  to  the  product  of  the  sum  of 
the  extreme  ordinates,  four  times  the  sum  of  the  even-num- 


198  GEOMETRICAL  APPLICATIONS.  [Art.  I7O0 

bered  ordinates,  and  twice  the  sum  of  the  remaining  odd-num- 
bered ordinates,  multipHed  by  one  third  of  the  common  interval. 
Again,  if  the  base  be  divided  into  a  number  of  parts  divis- 
ible by  three,  we  have,  by  applying  the  formula  derived  in 
Art.  169,  to  the  areas  between  the  ordinates  ^^1:^4,^4/7,  and  so  on, 

^  "^  V  ^^'  "^  ^-^^  "^  ^^^  +  274  +  3J/5  .  .  .   +  3J«  +  Jn+x). 

Cotes   Method  of  Approximation, 

171.  The  method  employed  in  Articles  168  and  169  is 
known  as  Cotes  Method.  It  consists  in  assuming  the  given 
curve  to  be  a  parabolic  curve  of  the  highest  order  which  can 
be  made  to  pass  through  the  extremities  of  a  series  of  equi- 
distant measured  ordinates. 

The  equation  of  the  parabolic  curve  of  the  ;^th  order  con- 
tains n  ■\-  \  unknown  constants;  hence,  in  order  to  eliminate 
these  constants  from  the  expression  for  an  area  defined  by  the 
curve,  it  is  in  general  necessary  to  have  n  +  i  equations  con- 
necting them  with  the  measured  ordinates.  Hence,  if  n  de- 
note the  number  of  intervals  between  measured  ordinates  over 
which  the  curve  extends,  the  curve  will  in  general  be  of  the 
n\\\  degree.* 

*  \i  H  denotes  the  whole  base,  the  first  factor  is  always  equivalent  to  H 
divided  by  the  sum  of  the  coefficients  of  the  ordinates  ;  for  if  all  the  ordinates  are 
made  equal,  the  expression  must  reduce  to  Hy^.  Thus,  each  of  the  rules  for  an 
approximate  area,  including  those  derived  by  repeated  applications,  as  in  Art.  170, 
may  be  regarded  as  giving  an  expression  for  the  mean  ordinate.  The  coefficients 
of  the  ordinates,  according  to  Cotes'  method,  for  all  values  of  «  up  to  w  =  10,  may 
be  found  in  Bertrand's  Calcul  Integral,  pages  333  and  334.  For  example  (using 
detached  coefficients  for  brevity),  we  have,  when  «  =  4, 

-4  =  —  [7, 32,  12,  32,  7] ; 
and  when  «  =  6, 

TT 

A  ■=  - —  [41,  216,  27,  272,  27,  216,  41]. 
040 


§  XIII.]  THE   FIVE-EIGHT  RULE.  I99 

172.  For  example,  let  it  be  required  to  determine  the  area 
between  the  ordinates  yi  and  j/2,  in  terms  of  the  three  equi- 
distant ordinates/i,  Jo  and_)/3,  the  common  interval  being  h. 

We  must  assume 

y=  A  ^-  Bx  -v  C^\ 

then  taking  the  origin  at  the  foot  of  ji, 

A=\yd.=  hlA^-^-\^ 

from  which  A,  B  and  C  must  be  eliminated  by  means  of  the 
equations 

yQ  =  A  +  2Bh  +  ^Cl^, 
Solving  these  equations,  we  obtain 

2 

If  we  make  a  slight  modification  in  the  ratios  of  these  last  coefficients  by  sub*, 
stituting  for  each  the  nearest  multiple  of  42,  we  have 

A  —  - —  [42,  210,  42,  252,  42,  210,  42], 
840 

(the  denominator  remaining  unchanged,  since  the  sum  of  the  coefficients  is  still 
840),  which  reduces  to 

IT 

^  =  —  [i,  5,  I,  6,  I,  5,  ij. 

This  result  is  known  as  Weddles  Rule  for  six  intervals.  The  vallie  thus  given  to 
the  mean  ordinate  is  evidently  a  very  close  approximation  to  that  resulting,  from 
Cotes'  method,  the  difference  being 

840  l^^i  +  ■^"  +  ^5  {y-i  +  Jo)  -  6  {yi  +  jKe)  -  20^4]. 


200  GEOMETRICAL   APPLICATIONS.  [Art.   1 72. 


and  substituting 

h 

173.  It  is,  however,  to  be  noticed,  that  when  the  ordinates 
are  symmetrically  situated  with  respect  to  the  area,  if  n  is 
everiy  the  parabolic  curve  may  be  assumed  of  the  {n  +  i)th 
degree.  For  example,  in  Art.  168,  71  —  2,  but  the  curve  was 
assumed  of  the  third  degree.  Inasmuch  as  A,  B,  C  and  D 
cannot  all  be  expressed  in  terms  of  j/j,  y^,  and  y^,  we  see  that  a 
variety  of  parabolic  curves  of  the  third  degree  can  be  passed 
through  the  extremities  of  the  measured  ordinates,  but  all  of 
these  curves  have  the  same  area."^ 

Application  to  Solids, 

(74.  \i  y  denotes  the  area  of  the  section  of  a  solid  perpen- 
dicular to  the  axis  of  x,  the  volume  of  the  solid  is  \ydx,  and 

*  This  circumstance  indicates  a  probable  advantage  in  making  n  an  even  num- 
ber when  repeated  applications  of  the  rules  are  made.  Thus,  in  the  case  of  six 
intervals,  we  can  make  three  applications  of  Simpson's  first  rule,  giving 

TT 

A  =  -  -  [i,  4,  2,  4,  2,  4,  i], (i) 

10 

or  two  of  Simpson's  second  rule,  giving 

'■^  =  ^  [i,  3,  3,  2,  3,  3,  i] (2) 

In  the  first  case,  we  assume  the  curve  to  consist  of  three  arcs  of  the  third  degree, 
meeting  at  the  extremities  of  the  ordinates  ^3  and  jr,  ;  but,  since  each  of  these  arcs 
contains  an  undetermined  constant,  we  can  assume  them  to  have  common  tangents 
at  the  points  of  meeting.  We  have  therefore  a  smooth,  though  not'  a  continuous 
curve.  In  the  second  case,  we  have  two  arcs  of  the  third  degree  containing  no 
arbitrary  constants,  and  therefore  making  an  angle  at  the  extremity  of  jj/4.  It  is 
probable,  therefore,  that  the  smooth  curve  of  the  first  case  will  in  most  cases  form  a 
better  approximation  than  the  broken  curve  of  the  second  case. 

In  confirmation  of  this  conclusion,  it  will  be  noticed  that  the  ratios  of  the 
coefficients  in  equation  (i)  are  nearer  to  those  of  Cotes'  coefficients  for  «  =  6,  given 
in  the  preceding  foot-note,  than  are  those  in  equation  (2). 


§  XIII.l 


APPLICATION   TO   SOLIDS. 


20I 


therefore  the  approximate  rules  deduced  in  the  preceding  arti- 
cles apply  to  solids  as  well  as  to  areas.  Indeed,  they  may  be 
applied  to  the  approximate  computation  of  any  integral,  by 
putting  J/  equal  to  the  coefficient  of  x  under  the  integral  sign. 

The  areas  of  the  sections  may  of  course  be  computed  by 
the  approximate  rules. 


Woolley's  Rule, 

175.  When  the  base  of  the  solid  is  rectangular,  and  the 
ordinates  of  the  sections  necessary  to  the  application  of  Simp- 
son's first  rule  are  measured,  we  may,  instead  of  applying  that 
rule,  introduce  the  ordinates  directly  into  the  expression  for 
the  area  in  the  following  manner. 

Taking  the  plane  of  the  base  for  the  plane  of  xy^  and  its 
centre  for  the  origin,  let  the  equation  of  the  upper  surface  be 
assumed  of  the  form 

z=A  ^Bx^Cy^D.^^Exy^Ff-vGj(^^Hx''y\-Ixf^Jf. 

Let  2h  and  2k  be  the  dimensions  of  the  base,  and  denote 
the  measured  values  of  z  as  indicated  in 
Fig.  25.     The  required  volume  is  '^ 


=  1       \  ^dy 
)  -h  i  -k 


dx. 


This  double  integral  vanishes  for  every 
term  containing  an  odd  power  of  x  or  an 
odd  power  oi  y\  hence 


hh 

=  — [12^  +  4i;>^  +  4^>^]. 


(I) 


202  GEOMETRICAL  APPLICATIONS.  [Art.  1 75. 

By  substituting  the  values  of  x  and  y  in  the  equation  of  the 
surface,  we  readily  obtain 

b2  =  A, (2) 

^1  +  <3:3  +  ^1  +  ^s  =  4^  +  4Dh^  +  aF]^,     ...    (3) 

«2  +  ^2  +  '^i  +  <^3  =  4^  +  '2'Dh^  +  2FB,  ...    (4) 

From  these  equations  two  very  simple  expressions  for  the 
volume  may  be  derived ;  for,  employing  (2)  and  (4),  equation 
(i)  becomes 

^=^(^  +  ^1  +  2^2  +  ^3+^2);     .     .     .     .  (4) 

and  employing  (2)  and  (3), 

hk 
F=  — -  (^1  +  ^  +  8/^2  +  <^i  +  ^^s) (5) 

Equation  (4)  is  known  as  Woolleys  Rule ;  the  ordinates  employed 
are  those  at  the  middles  of  the  sides  and  at  the  centre  ;  in  (5), 
they  are  at  the  corners  and  at  the  centre. 


Examples  XI 11. 

1.  Apply  Simpson's  Rule  to  the  sphere,  the  hemisphere,  and  the 
cone,  and  explain  why  the  results  are  perfectly  accurate. 

2.  Apply  Simpson's  Second  Rule  to  the  larger  segment  of  a  sphere 
made  by  a  plane  bisecting  at  right  angles  a  radius  of  the  sphere. 

~8~' 


§  XIII.]  EXAMPLES.  203 

3.  Find  by  Simpson's  Rule  the  volume  of  a  segment  of  a  sphere,  b 
and  c  being  the  radii  of  the  bases,  and  h  the  altitude. 

4.  Find  by  Simpson's  Rule  the  volume  of  the  frustum  of  a  cone,  b 
and  c  being  the  radii  of  the  bases,  and  h  the  altitude. 

—  {lfi^-bc-\-  c"). 

5.  Compute  by  Simpson's  First  and  Second  Rules,  the  value  of 

,  the  common  interval  being  ^V  in  each  case. 

oi  +  ^  ^  ^'' 

The  first  rule  gives  0.6931487,  and  the  second  rule  gives  0.6931505. 

The  correct  value  is  obviously  loge2  =  0.6931472. 

6.  Find  the  volume  considered  in  Art.  175,  directly  by  Simpson's 
Rule,  and  show  that  the  result  is  consistent  with  equations  (4)  and  (5). 

hk 
y=  —  [ai  -{-  as  -h  Ci  +  d  -{-  4  {a<i  -h  bi  +bi  -\-  c^)  4- 16^2]. 

7.  Find,  by  elimination,  from  equations   (4)  and  (5),  Art.  175,  a 
formula  which  can  be  used  when  the  centre  ordinate  is  unknown. 

V=  —  [4(^3  +  b^  +  bi  -f  (Ta)  —  {a^  +  ai  +  Ci  +  Ct)\. 
o 


204  MECHANICAL   APPLICATIONS.  [Art.  1 76. 

CHAPTER  IV. 

Mechanical  Applications. 


XIV. 
Definitions. 

J76.  We  shall  give  in  this  chapter  a  few  of  the  applications 
of  the  Integral  Calculus  to  mechanical  questions. 

The  mass  or  quantity  of  matter  contained  in  a  body  is  pro- 
portional to  its  weight.  When  the  masses  of  all  parts  of  equal 
volume  are  equal,  the  body  is  said  to  be  homogeneous.  The 
factor  by  which  it  is  necessary  to  multiply  the  unit  of  volume 
to  produce  the  unit  of  mass  is  called  the  density^  and  usually 
denoted  by  y. 

In  the  following  articles  it  will  be  assumed,  when  not  other- 
wise stated,  that  the  body  is  homogeneous,  and  that  the  density 
is  equal  to  unity,  so  that  the  unit  of  mass  is  identical  with  the 
unit  of  volume.  When  the  mass  of  an  area  is  spoken  of,  it  is 
regarded  as  a  lamina  of  uniform  thickness  and  density,  and  the 
unit  of  mass  is  taken  to  correspond  with  the  unit  of  surface. 
In  like  manner  the  unit  of  mass  for  a  line  is  taken  as  identical 
with  the  unit  of  length. 

Statical  Moments, 

177.  The  moment  of  a  force,  with  reference  to  a  point,  is  the 
measure  of  the  effectiveness  of  the  force  in  producing  motion 
about  the  point.  It  is  shown  in  treatises  on  Mechanics,  that 
this  is  the  product  of  the  force  and  the  perpendicular  from  the 
point  upon  the  li?ie  of  application  of  the  force. 


§  XIV.]  STATICAL  MOMENTS.  205 

The  moment  of  the  sum  of  a  number  of  forces  about  a 
given  point  is  the  sum  of  the  moments  of  the  forces. 

The  statical  inornent  of  a  body  about  a  given  point  is  the 
moment  of  its  gravity  ;  the  force  of  gravity  being  supposed  to 
act  upon  every  part  of  the  body,  and  in  parallel  lines. 

178.  In  order  to  find  the  statical  moment  of  a  continuous 
body,  we  regard  the  body  as  generated  geometrically  in  some 
convenient  manner,  and  determine  the  corresponding  differen- 
tial of  the  moment. 

In  the  case  of  a  plane  area,  let  the  body  be  referred  to 
rectangular  axes,  and  let  gravity  be  supposed  to  act  in  the 
direction  of  the  axis  oi  y.  Then  the  abscissa  of  the  point  of 
application  is  the  arm  of  the  force  when  we  consider  the 
moment  about  the  origin.  Let  us  first  suppose  the  area  to  be 
generated  by  the  motion  of  the  ordinate  y.  The  differential  of 
the  area  is  then  y  dx.     The  corresponding  element  of  the  sum, 

of  which  the  integral    y  dx  is  the  limiting  value,  see  Art.  99,  is 

i  a 

yr^x, (l) 

in  which  jjv  is  the  ordinate  corresponding  to  any  value  of  x 
intermediate  between  a  -h  (r  —  1)  ax,  and  a  -\-  r  Ax.  It  is 
evident  that  the  arm  of  the  weight  of  the  element  (i)  is  such 
an  intermediate  value  of  x  ;  hence  the  moment  of  the  ele- 
ment is 

x^yr  Ax. (2) 

The  whole  moment  is  therefore  the  limiting  value  of  a  sum 
of  the  form 

^\xryr  ^x. 


In  other  words,  it  is  the  integral 


xy  dx, 


(3) 


206 


MECHANICAL  AP PLICA  TIONS. 


[Art.  178. 


in  which  the  differential  of  the  moment  is  the  product  of  the 
differential  of  the  area  and  the  arm  of  the  force,  which  in  this 
case  is  the  same  for  every  point  of  the  element.  In  other 
words,  the  moment  of  the  differential  is  the  differential  of  the 
moment. 

179.  As  an  illustration,  we  find  the  moment  of  a  semicircle 
(Fig.  26)  about  its  centre.  The  area  may  be 
generated  by  the  line  2r,  moving  from  ;t'  =  o  to 

Y 


X  =  a.     The  equation  of  the  circle  being 

x"  ^f  =  d\ 


the  differential  of  the  area  is 


Fig.  26. 


2  4/(<^-  —  .r^)  dx. 
The  moment  of  this  differential  is 
2  \/{c^  —  :^\x  dx ; 
hence  the  whole  moment  is 

2  r  V(«2  -  ^P^x  dx  =  ^'^-  (a'  -  jt^fT  =  — 

'  o  0  _lo  ^ 


Centres  of  Gravity, 

180.  If  a  force  equal  to  the  whole  weight  of  a  body  be 
applied  with  an  arm  properly  determined,  its  moment  may  be 
made  equivalent  to  the  whole  statical  moment  of  the  body. 
If  the  force  is  in  the  direction  of  the  axis  of  y,  as  in  Fig.  26,  we 
have,  denoting  this  arm  by  "x, 

Ic  •  Area  =  Moment, 
Moment 


X  — 


Area 


§  XIV.]  CENTRES  OF  GRA  VITY.  20/ 

In  like  manner,  supposing  the  force  to  act  in  the  direction 
of  the  axis  of  x^  we  may  determine  y  for  the  same  body. 

It  is  shown  in  treatises  on  Mechanics  that  the  point  deter- 
mined by  the  tWo  coordinates  x  and  y,  is  independent  of  the 
position  of  the  coordinate  axis.  This  point  is  called  the  centre 
of  gravity  of  the  area.  The  centre  of  gravity  of  a  volume  is 
defined  in  like  manner. 

181.  The  symmetry  of  the  form  of  a  body  may  determine 
one  oi  more  of  the  coordinates  of  its  centre  of  gravity.  Thus 
the  centre  of  gravity  of  a  circle  or  a  sphere  coincides  with  the 
geometrical  centre,  and  the  centre  of  gravity  of  a  solid  of  revolu- 
tion is  on  the  axis  of  revolution.  The  centre  of  gravity  of  the 
semicircle  in  Fig.  26,  is  on  the  axis  of  x\  hence  to  determine 
its  position  we  have  only  to  find  'x.  Dividing  the  moment 
of  the  semicircle  found  in  Art.  179  by  the  area  \nc?^  we  have 

_      Aa 

182.  In  finding  the  moment  of  the  semicircle  (Art.  179),  we 
regarded  the  area  as  generated  by  the  double  ordinate  27,  and 
the  differential  of  the  moment  was  found  by  multiplying  the 
differential  of  the  area  by  x,  which  is  the  arm  of  the  force  for 
every  point  of  the  generating  line. 

We  may,  however,  derive  the  moment  from  the  differential 
of  area, 

■^^7, (0 

since  the  area  may  be  generated  by  the  motion  of  the  abscissa 
X  from  y=  —  a  to  y  =  a.  But  in  this  case  to  find  the  moment 
of  the  differential  we  must  multiply  it  by  the  distance  of  its 
centre  of  gravity  from  the  given  axis.  The  centre  of  gravity  of 
the  line  x  is  evidently  its  middle  point,  hence  the  required  arm 
is  Ix.     Therefore  the  differential  of  the  moment  is 

.       ^; (2) 


208  MECHANICAL   APPLICATIONS.  [Art.  1 82. 

and  consequently  the  whole  moment  is 

^  J  ~a  --  J  -a  O 

This  result  is  identical  with  that  derived  in  Art.  179. 

Polar  Formulas, 

183.  When  polar  formulas  are  employed,  r  and  B  being 
coordinates  of  the  curved  boundary  of  the  area,  the  element  is 
\7^  dS.  Since  this  element  is  ultimately  a  triangle,  we  employ 
the  well  known  property  of  triangles  ;  that  the  centre  of  gravity 
is  on  a  medial  line  at  two-thirds  the  distance  from  the  vertex 
to  the  base. 

The  coordinates  of  the  centre  of  gravity  of  the  element  are, 
therefore, 


2  2 

-rsin6'  and  -rcosB. 


Hence  we  have  the  formula 


dO 


Ur  cos  d^f^dO      ^    [f^cosO 

iir'de  ^         [r^dO     ' 

[7^  sine  df^ 
and  similarly  y  =  -  .  — • 

^        jr^  dd 


§  XIV.]  '  POLAR  FORMULAS.  209 

(84-.  To  illustrate,  let  us  find  the  centre  of  gravity  of  the 
area  enclosed  by  the  lemniscata 

7^  —  a^  cos  20. 


Whence     x  =z  — 


[  {cos  26  f  COS  Odd  z 

^-^ — =^y  (COS 2d)  COS dde. 

r~  3    Jo 

I: 


COS  26  do 


Put  COS  26  —  cos^  ^,         whence  sin  (j)  =  \/2  sin  6j 

and  V2  cos  B  dB  —  cos  <i>  d(j)y 


-        2V2       f 2  .,    ,,        ^  ,  ^      J      .     .. 

X  =■ a      cos*(p  d(f)  — (7  =  -TT-  Tta. 

3      •'o 


24/2      3-1      TV      _    V2 


Solids  of  Revolution, 

185.  To  find  the  centre  of  gravity  of  a  solid  of  revolution, 
we  take  the  axis  of  revolution  as  the  axis  of  x^  and  the  circle 
whose  area  is  nf'  as  the  generating  element.  Replacing  y  in 
equation  (3),  Art.  178,  by  this  expression,  we  have  for  the  stati- 
cal moment 

7t     xf"  dx, 

and  for  the  abscissa  of  the  centre  of  gravity 

_          xfdx 
X  —  ^  "^ 

dx 


i  a 


210  MECHANICAL  APPLICATIONS.  [Art.  1 86. 

186.  To  illustrate,  we  find  the  centre  of  gravity  of  a  spheri- 
cal segment  whose  height  is  //.  In  this  case,  taking  the  origin 
at  the  vertex  of  the  segment,  and  denoting  the  radius  of  the 
sphere  by  a^  we  have 

f-  —  2ax  —  :^. 

fh  2  I      "l'^ 

{2ax'  —  ^)  dx      -a^  -  -x^  \  ,    ^ 

Hence       x  =  h ^3  A    ^q   ^h   ^a  -  ^h ^ 

f'  (2ax  -  x^)  dx       a^  -  -^t-^T       4    3^-^* 
Jo  3     Jo 

If  the  centre  of  gravity  of  the  surface  of  the  segment  be  re- 
quired, since  the  differential  of  the  surface  is  27ty  ds,  we  easily 
obtain  the  general  formula 


x  = 


and,  in  this  case  the  curve  being  a  circle,  y  ds  =  a  dx]  hence, 
substituting,  we  have 

X  =  ^h. 


The  Properties  of  Pappus, 

187.  Let  a  solid  be  generated  by  the  revolution  of  any  plane 
figure  about  an  exterior  axis  in  its  own  plane.  It  is  required 
to  determine  the  volume  and  the  surface  thus  generated. 

It  is  evident  that  this  solid  may  also  be  generated  by  a 
variable  circular  ring  whose  centre  moves  along  the  axis  of 
revolution  ;  denoting  by  jj  and  72  corresponding  ordinates  of 


§  XIV.]  THE  PROPERTIES  OF  PAPPUS.  211 

the  outer  and  inner  circles  respectively,  the  area  of  this  ring  is 
7i{yi  —  yi).     Hence 

But   this  integral  is  the  statical  moment  of  the  given  figure, 

since /i  —  y^  is  the  generating  element  of  its  area,  and  — — ^is 

the,  corresponding  arm.  Denoting  the  area  of  the  figure  by  Ay 
we  may  therefore  write 

V=  2nyA  ; 

that  is,  the  volume  is  the  product  of  the  area  of  the  figure  and  the 
path  described  by  its  centre  of  gravity. 

The  surface  (5)  of  this  solid  is,  by  Art.  149, 

S  —  27t\yds  =27t\dSf 
if  J  denotes  the  ordinate  of  the  centre  of  gravity  of  the  arc  s. 

Hence  we  have  S=  ZTrj-arc ; 

that  is,  the  surface  is  the  product  of  the  length  of  the  arc  into 
the  path  described  by  the  centre  of  gravity. 

These  theorems  are  frequently  called  the  properties  of  Gul- 
dinus ;  they  are,  however,  due  to  Pappus,  who  published  them 
1588. 

It  is  obvious  that  both  theorems  are  true  for  any  part  of 
a  revolution  of  the  generating  figure. 


212  MECHANICAL  APPLICATIONS.  [Ex.  XIV, 


Examples    XIV. 

1.  Find  the  centre  of  gravity  of  the  area  enclosed  between  the 
parabola  y^  =  ^mx  and  the  double  ordinate  corresponding  to  the 
abscissa  a. 

5 

2.  Find  the  centre  of  gravity  of  the  area  between  the  semi-cubical 
parabola  af  =  x^  and  the  double  ordinate  which  corresponds  to  the 
abscissa  a. 

7 

3.  Find  the  ordinate  of  the  centre  of  gravity  of  the  area  between 
the  axis  of  x  and  the  sinusoid  y  =  sin  .r,  the  limits  being  x  =  o  and 
x=7t.  y=i7r. 

4.  Find  the  coordinates  of  the  centre  of  gravity  of  the  area  be- 
tween the  axes  and  the  parabola 


ey-©*- 


X  =  —  ,  and  y  =  - 
5  5 


5.  Find  the    centre   of  gravity  of  the  area  between    the   cissoid 
f  {a  —  x)  —  x^  and  its  asymptote. 
Solution  : — 

Denoting  the  statical  moment  by  M  and  the  area  by  A, 

M  =       =  —  2x-^  (a  —  xY      +5      ^^"^  {^  —  -^'f^  ^x 

Jo  {a  —  X)-'-  Jo         Jo 

=  z^a-  A-  sM; 

,'.M-=^A,  hence  a=^. 

0  o 


§  XIV.]  EXAMPLES.  213 

6.  Find  the  centre  of  gravity  of   the  area  between  the  parabola 

v'  =  ^ax  and  the  straight  line  j  ~  mx. 

—        %a  .  -      211 

X  ■=  — :, ,  and  y  —  — . 
5w  m 

7.  Find  the  centre  of  gravity  of  the  segment  of  an  ellipse  cut  off 
by  a  quadrantal  chord. 

—       2a  ,    —       2         b 

X  =  -  • ,  and  y 


$    7r  —  2  -       s     TT-  2 

8.  Given  the  cycloid, 

y  —  a{i  —  cos ?/.'),  X  =  a  (ip  —  sin  ip) , 

find  the  distance  of  its  centre  of  gravity  from  the  base. 

^  =  6- 

9.  Find  the  centre  of  gravity  of  the  area  enclosed  between  the 
positive  directions  of  the  coordinate  axes  and  the  four-cusped  hypo- 
cycloid 

x^  -\-  y^  =  a^. 

Put  .\"  =  «  cos^  0,  and y  =  a  sin'  0. 

-  3^^ 


x=y 


10.  Find  the  centre  of  gravity  of  the  area  enclosed  by  the  cardioid 
.   =  a(i  —  cos  0). 


^=-f 


II.  Find  the  centre  of  gravity  of  the  sector  of  a  circle  whose  radius 

is  a,  the  angle  of  the  sector  being  2  a. 

—        2  a  sin  (X 
Use  the  method  of  Art.  i^2>-  '*' 


3 


a 


214  MECHANICAL  APPLICATIONS.  [Ex.  XIV. 

12.  Find  the  centre  of  gravity  of  the  segment  of  a  circle,  the  angle 
subtended  being  2  a  and  the  radius  of  the  circle  a. 


Solution 


X 


J  a  cos  a 


2  1       \a  —X  )  xdx  3.3  3 

2a  sin  a       Chord 


Area  3  Area         12  Area 


13.  Find  the  centre  of  gravity  of  a  circular  ring,  the  radii  being  a 
and  «i,  and  the  angle  subtended  2a. 

-  _  2    d  —  a^     sin  (y 

3    d^  —  a-c       oc 

14.  Find  the  centre  of  gravity  of  a  circular  arc,  whose  length  is  2s. 

Soliction : — 

We  have  in  this  case,  taking  the  origin  at  the  centre  and  the  axis 
of  X  bisecting  the  arc, 


xds 

X 


ds 


Put  X  —  a  cos  0,  then  ds  —  a  dB,  and  denoting  by  a  the 

angle  subtended  by  ^,  we  have 


fOL 

X  =   


COS  0  do 

^  sin  «        c 


2s  a  a 

2c  being  the  chord. 


§  XIV.]  EXAMPLES.  215 

15.  Find  the  coordinates  of  the  centre  of  gravity  of  arc  of  the  semi- 
cycloid  whose  equations,  referred  to  the  vertex,  are 

;r  =  «  (i  —  cos  ^'),  and  j  ==  « (^  +  sin  ^). 

^^,  andj^r^  [n-Yja- 


X 


16.  Find  the  centre  of  gravity  of  the  arc  between  two  successive 
cusps  of  the  four-cusped  hypocycloid 


x^  4-7^  =  a^\ 


_  _  _  _  2d! 


17.  Find  the  position  of  the  centre  of  gravity  of  the  arc  of  the  semi- 
cardioid 

r  =  «  (i  —  cos  6). 

x= ,  and  y  =  —  . 

18.  A  semi-ellipsoid  is  formed  by  the  revolution  of  a  semi-ellipse 
about  its  major  axis  ;  find  the  distance  of  the  centre  of  gravity  of  the 
solid  from  the  centre  of  the  ellipse. 

x-^ 

19.  Find  the  centre  of  gravity  of  a  frustum  of  a  paraboloid  of 
revolution  having  a  single  base,,  k  denoting  the  height  of  the  frustum. 

'*^~   3  ■ 

20.  A  paraboloid  and  a  cone  have  a  common  base  and  vertices  at 
the  same  point  ;  find  the  centre  of  gravity  of  the  solid  enclosed 
between  them. 

The  centre  of  gravity  is  the  middle  point  of  the  axis. 


2l6  MECHANICAL  APPLICATIONS,  [Ex.  XIV. 

21.  Find  the  centre  of  gravity  of  a  hyperboloid  whose  height  is  hy 
the  generating  curve  being 

y^  —  m  (2ax  +  ^'). 

—  k    Sa  +  s^ 
x  = V  . 

4     s^-\-  k 

22.  Find  the  centre  of  gravity  of  the  solid  formed  by  the  revolution 
of  the  sector  of  a  circle  about  one  of  its  extreme  radii. 

The  height  of  the  cone  being  denoted  by  ^,  and  the  radius  of  the 
circle  by  «,  we  have 

23.  Find  the  centre  of  gravity  of  the  solid  formed  by  the  revolution 
about  the  axis  of  x  of  the  curve 

ay  =  ax^  —  x^f 

between  the  limits  o  and  a. 

x-^ 

24.  A  solid  is  formed  by  revolving  about  its  axis  the  cardioid 

r  —  a  (i  —  cosG)  ; 

find  the  distance  of  the  cusp  from  the  centre  of  gravity. 

—  _  i6a 

25.  Determine  the  position  of  the  centre  of  gravity  of  the  volume 
included  between  the  surfaces  generated  by  revolving  about  the  axis 
of  .;*:  the  two  parabolas 

y  =  mxy  and  y^  =  m'  {a  —  x). 

-  a    m  +  2m' 

X 


3     m  +  m 


§  XIV.]  EXAMPLES,  217 

26.  Find  the  centre  of  gravity  of  a  rifle  bullet  consisting  of  a  cylin- 
der two  calibers  in  length,  and  a  paraboloid  one  and  a  half  calibers  in 
length  having  a  common  base,  the  opposite  end  of  the  cylinder  con- 
taining a  conical  cavity  one  caliber  in  depth  with  a  base  equal  in  size 
to  that  of  the  cylinder. 

The  distance  of  the  centre  of  gravity  from  the  base  of 
the  bullet  is  if  I  calibers. 

27.  A  solid  formed  by  the  revolution  of  a  circular  segment  about 
its  chord  is  cut  in  halves  by  a  plane  perpendicular  to  the  chord  ; 
determine  the  centre  of  gravity  of  one  of  the  halves.  This  solid  is 
called  an  ogival. 

Denoting  hy  2a  the  angle  subtended  by  the  chord,  and  by  a  the 
radius  of  the  circle,  the  distance  of  the  centre  of  gravity  from  the 
base  is 

-  _  a     44  sin*^  a  +  sin'  2a  +  32  (cos  201  —  cos  a) 


X  = 


6  sin  Of  (2  -I-  cos"  a)  —  ^a  cos  a 


28.  Find  the  centre  of   gravity  of  the  surface  of  the  paraboloid 
formed  by  the  revolution  about  the  axis  of  x  of  the  parabola 


/  =  4mx, 
a  denoting  the  height  of  the  paraboloid. 

-  _  I     (3^  ~  2»2)  {a  +  m)^  -t-  2fn^ 

X  —  —  • — — , 

5  {a  -\-  my  —  m^ 

29.  Find  the  centre  of  gravity  of  the  surface  generated  by  the  revo- 
lution of  a  semi-cycloid  about  its  axis,  the  equations  of  the  curve 
being 

;i;  =  ^(i  _  cos  ^),  and  jj' =  d!  (^  +  sin  ^). 

-        2a     IKTT  —  8 

x  = ^ . 

15      3^-4 


2l8  MECHANICAL  APPLICATIONS.  [Ex.  XIV. 

30.  Find  the  centre  of  gravity  of  the  surface  generated  by  the  revo- 
lution about  its  axis  of  one  of  the  loops  of  the  lemniscata 

r^  =1  a^  cos  20. 

-      2  +  V2 
X  :=■  — -— a. 


31.  A  cardioid  revolves  about  its  axis  ;  find  the  centre  of  gravity 
of  the  surface  generated,  the  equation  of  the  cardioid  being 


r  =  a  {1  —  cos9)- 

63 


—      f^oa 


32.  A  ring  is  generated  by  the  revolution  of  a  circle  about  an  axis 
in  its  own  plane  ;  c  being  the  distance  of  the  centre  of  the  circle 
from  the  axis,  and  a  the  radius,  determine  the  volume  and  surface 
generated. 

V—  27t^cc^^  and  S—  ^n'^ca. 

33.  A  triangle  revolves  about  an  axis  in  its  plane  ;  ax,  a^,  and  a^^ 
denoting  the  distances  of  its  vertices  from  the  axis,  determine  the  vol- 
ume generated. 

271 A  ,  . 

V  — ■  \ax  +  ^2  +  ^3). 


34.  Find  the  Volume  of  a  frustum  of  a  cone,  the  radii  of  the  bases 
being  ax  and  a^^,  and  the  height  h, 

7th 


{ux  +  axa^  +  «/). 


35.  Find  the  volume  and  surface  generated  by  the  revolution  of  a 
cycloid  about  its  base. 

647ra^ 


V=  57rV,  and  S  = 


§  XV.]  MOMENTS  OF  INERTIA,  2ig 

XV. 

Moments  of  Inertia, 

188.    When  a  body  rotates  about  a  fixed  axis,  the  velocity 
of  a  particle  at  a  distance  r  from  the  axis  is 

in  which  go  is  the  angle  of  rotation.  The  force  which  acting 
for  a  unit  of  time  would  produce  this  motion  in  a  mass  m  is 
measured  by  the  momentum 

daa 

mr  —r- . 

The  moment  of  this  force  about  the  axis  is  therefore 

o  ddj 
m'T  —T" 
dt 

The  sum  of  these  moments  for  all  the  parts  of  a  rigid  system  is 

since  the  angular  velocity,  -5- ,  is  constant.      In  the  case  of  a 

dt 

continuous  body  this  expression  becomes 

in  which  dm  is  the  differential  of  the  mass.     The  factor 

\r^dm^ 


220  MECHANICAL  APPLICATIONS.  [Art.   1 88. 

which  depends  upon  the  shape  of  the  body,  is  called  its  mo^ 
ment  of  inertia^  and  is  denoted  by  /. 

189.  When  the  body  is  homogeneous,  dm  is  to  be  taken 
equal  to  the  differential  of  the  line,  area,  or  volume,  as  the  case 
may  be.  For  example,  in  finding  the  moment  of  inertia  of  a 
straight  line  whose  length  is  2a^  about  an  axis  bisecting  it  at 
right  angles,  we  let  x  denote  the  distance  of  any  point  from 
the  axis;  then  dm  =  dx,  hence  we  have 


I^l'  x^dx  =  ^-^=^^ 

J-.  ^  12 


Again,  in  finding  the  moment  of  inertia  of  the  semi-circle  in 
figure  25,  about  the  axis  of  j^,  let  dm=  2ydx\  then,  since  every 
point  of  the  generating  line  is  at  the  distance  x  from  the  axis, 
the  moment  of  inertia  is 

7=2     yx^  dx  =  2\     V{a^  —  ^)  ^^  dx , 

Jo  Jo 

Putting  X  —  a  sin  6,  we  have 


1=  20^  f'  cos^  e  sin2  ede  =  ^\ 

Jo  O 


T/^e  Radius  of  Gyration, 

190.  If  the  whole  mass  of  the  body  were  situated  at  the 
distance  k  from  the  axis,  its  moment  of  inertia  would  be  Bm. 
Now,  if  k  is  so  determined  that  tJns  moment  shall  be  equal  to 
the  actual  moment  of  inertia  of  the  body,  the  value  of  k  is  the 
radius  of  gyration  of  the  body  with  reference  to  the  given 
axis.     Hence 

j^  __  Moment  of  inertia 
Mass  * 


§  XV.]  THE   RADIUS  OF  GYRATION.  221 

Thus,  for  the  radius  of  gyration  of  the  line  2a^  whose  moment 
of  inertia  is  found  in  the  preceding  article,  we  have 

>r=- ,  or  k=  —\ 

3  V3 

and  for  the  radius  of  gyration  of  the  semi-circle,  whose  area 
is  \7ia^y 

J^^""-,  or  k^""-. 

4  2 

It  is  evident  that  this  expression  is  also  the  radius  of  gyra- 
tion of  the  whole  circle  about  a  diameter,  for  the  moment  of 
inertia  of  the  circle  is  evidently  double  that  of  the  semi-circle, 
and  its  area  is  also  double  that  of  the  semi-circle. 

191.  It  is  sometimes  convenient  to  use  modes  of  generating 
the  area  or  volume,  other  than  those  involving  rectangular 
coordinates.  For  example,  let  it  be  required  to  find  the  radius 
of  gyration  of  a  circle  whose  radius  is  a^  about  an  axis  passing 
through  its  centre  and  perpendicular  to  its  plane.  This  circle 
may  be  generated  by  the  circumference  of  a  variable  circle 
whose  radius  is  r,  while  r  passes  from  o  to  a.  The  differential 
of  the  area  is  then  2nr  dr,  and  the  moment  is 


I  =  27t\    7^  dr  =  —  , 


'i: 

Dividing  by  the  area  of  the  circle,  we  have 


192.  Again,  to  find  the  radius  of  gyration  of  a  sphere 
whose  radius  is  a  about  a  diameter.  In  order  that  all  points 
of  the  elements  shall  be  at  the  same  distance  from  the  axis. 


222  MECHANICAL  APPLICATIONS.  [Art.  I92. 

we  regard  the  sphere  as  generated  by  the  surface  of  a  cylinder 
whose  radius  is  Xy  and  whose  altitude  is  2y.  The  surface  of 
this  cylinder  is  therefore  A^rcxy.  The  differential  of  the  volume 
\s>  ^Ttxy  dx^  and  the  moment  of  inertia  is 


I  —  ^n  \x^y  dx  =  47c     ^{cp-  —  x)  x^  dx. 
Putting X  =  asAn  0^ 


I  =  47ra'  f '  sin^  6  cos^^  0  dO  =  ^ . 
Jo  15 


Dividing  by  ~ —  ,  the  volume  of  the  sphere,  we  have 


^  =  ^', 


Radii  of  Gyration  about  Parallel  Axes, 

193.  The  moment  of  inertia  of  a  body  about  any  axis  exceeds 
its  moment  of  inertia  about  a  parallel  axis  passing  through  the 
centre  of  gravity^  by  the  product  of  the  mass  and  the  square  of 
the  distance  between  the  axes. 

Let  h  be  the  distance  between  the  axes.  Pass  a  plane 
through  the  element  dm  perpendicular  to  the  axes,  and  let  r 
and  rx  be  the  distances  of  the  element  from  the  axes.  Then, 
r,  ^i,  and  //  form  a  triangle ;  let  d  be  the  angle  at  the  axis 
passing  through  the  centre  of  gravity,  then 


,2  _ 


r{  +  U^  —  2ri/f  cos  B (i) 


§  XV.]      RADII  OF  GYRATION  ABOUT  PARALLEL  AXES.       223 

The  moment  of  inertia  is  therefore 

rl  dm  +  l^m  —  2h\r^  cos  d  dm  .     .     .     (2j 


T^dm  = 


Now  Ti  and  6  are  the  polar  coordinates  of  dm^  in  the  plane 
which  is  passed  through  the  element;  hence  the  last  integral  in 
equation  (2)  is  equivalent  to 


-2/l\ 


X  dm. 


But     X  dm  is  the  statical  moment  of  the  body  about  the  axis 

passing  through  the  centre  of  gravity.  Now  from  the  defini- 
tion of  the  centre  of  gravity,  this  moment  is  zero  ;  hence^ 
equation  (2)  reduces  to 

7^  dm  =    r^  dm  +  Ihn  ......  (3 

Introducing  the  radii  of  gyration,  we  have  also 

J^  =  ki^B (4) 

194.  As  an  application  of  this  result,  we  shall  now  find  the 
moment  of  inertia  of  a  cone  whose  height  is  h,  and  the  radius 
of  whose  base  is  a,  about  an  axis  passing  through  its  vertex 
perpendicular  to  its  geometrical  axis.  Taking  the  origin  at 
the  vertex  of  the  cone,  the  axis  of  x  coincident  with  the  geo- 
metrical axis,  and  a  circle  perpendicular  to  this  axis  as  the 
generating  element,  we  have  for  the  area  of  this  element  ny^^ 
and   for  its  radius  of  gyration  about  a  diameter  parallel  to 

the  given  axis,  — . 
4 


224  MECHANICAL  APPLICATIONS.  [Art.  1 94. 

The  distance  between  these  axes  being  x,  the  proposition 
proved  in  the  preceding  article  gives  an  expression  for  the 
radius  of  gyration  of  the  element  about  the  given  axis ;  viz., 

x^  +  —  .     Replacing  r^,   in  the  general  expression  for  /  (Art. 

4 
188),  by  this  expression,  and  substituting  for  dm  the  differen- 
tial ny^  dx^  we  have 


I 

=  n\{x^^t^fdx, 

in  which  y  — 

ax 
'  li 

' 

Therefore 

1  = 

nd' 
If 

'!'(■ 

Jo     \ 

(' 

and  since 

V-  ^"'"'^  , 

^=.A(,.  +  4,.). 


To    find    the    square    of   the    radius   of   gyration    about   a 
parallel  axis  through  the  centre  of  gravity,  we  have 


To  find  the  moment  of  inertia  of  a  right  cone  about  its 
geometrical  axis  we  employ  the  same  generating  element  as 
before  ;  but  in  this  case  the  square  of  the  radius  of  gyration  is 


Hence 
2 


y 

"l\'''"%i'""- 


§  XV.]      RADII  OF  GYRATION  ABOUT  PARALLEL  AXES.         22$ 


therefore 


1= ,     whence     ^=^^—. 

lO  lO 


Polar  Moments  of  Inertia. 


195.  In  the  case  of  a  plane  area,  when  the  axis  of  rotation 
passes  through  the  origin,  we  have 

r^  =  ^  -\-  ^,  hence    r^  dm  =    (jv^  +  j^)  dm, 

therefore  /=  \:t^  dm  +  ly^dm; 

that  is,  tke  sum  of  the  moments  of  inertia  of  a  plane  area  about 
two  axes  in  its  own  plane  at  right  angles  "to  each  other  is  equal  to 
the  moment  of  inertia  about  an  axis  through  the  origin  perpendicu- 
lar to  the  plane.  /  in  the  above  equation  is  called  the  polar 
mome7tt  of  inertia. 

In  the  case  of  the  circle,  since  the  moment  is  the  same 
about  every  diameter,  the  polar  moment  is  twice  the  moment 
about  a  diameter ;  that  is,  denoting  the  former  by  //  and  the 
latter  by  /«,  we  have 


See  Art.  191, 


Examples  XV. 


I.  Find  the  radius  of  gyration  of  a  circular  arc  (2^)  about  a  radius 
passing  through  its  vertex. 


226  MECHANICAL  APPLICATIONS.  [Ex.  XV. 

Solution : — 

Taking  the  origin  at  the  centre,  and  the  axis  of  x  bisecting  the  arc, 
'     and  denoting  hy  2a  the  angle  subtended  by  2s,  we  have 


mJk'  =  ['  /  ds  =  a'  f"   sin'  B  dQ. 

^,^.V^_sin^\ 
2   \  2a   J 


m  =  2aa 


2.  Find  the  radius  of  gyration  of  the  same  arc  about  the  axis  of  y^ 
and  thence  about  a  perpendicular  axis  through  the  centre  of  the 
circle.  k  =  a. 

3.  Find  the  radius  of  gyration  of  the  same  arc  about  an  axis  through 
its  vertex  perpendicular  to  the  plane  of  the  circle. 

See  Ex.  XIV.,  14,  and  denote  by  c  the  subtending  chord. 


>e^^a\.-^-). 


4.  Find  the  moment  of  inertia  of  the  chord  of  a  circular  arc,  in 

terms  of  the  diameter  parallel  to  it,  and  its  angular  distance  from  this 

diameter. 

73 

See  Arts.  189  and  193.  /  = —  (3  cos  a  —  cos  ^a) . 

24 

5.  Find  the  radius  of  gyration  of  an  ellipse  about  an  axis  through 
its  centre  perpendicular  to  its  plane. 

Eind  the  radius  of  gyration  about  the  major  axis  and  about  the  minor 
axisy  and  apply  Art,  195. 

k'  =  i{a'  +  b'), 

6.  Find  the  radius  of  gyration  of  an  isosceles  triangle  about  a  per 
pendicular  let  fall  from  its  vertex  upon  the  base  (2b). 

6- 


§  XV.]  EXAMPLES.  227 

7.  Find  the  radius  of  gyration  about  the  axis  of  the  curve,  of  the 
area  enclosed  by  the  two  loops  of  the  lemniscata 


r^  —  a'  cos  29. 


^•  = -3(3^^-8). 


«.  Find  the  radius  of  gyration  of  a  right  triangle,  whose  sides  are  a 
and  b,  about  an  axis  through  its  centre  of  gravity  perpendicular  to  its 
plan^ 

18 

9.  Find  the  radius  of  gyration  of  a  portion  of  a  parabola  bounded 
by  a  double  ordinate  perpendicular  to  the  axis,  about  a  perpendicular 
to  its  plane  passing  through  its  vertex. 

10.  Find  the  radius  of  gyration  of  a  cylinder  about  a  perpendicular 
that  bisects  its  geometrical  axis,  2/  being  the  length  of  the  cylinder, 
and  a  the  radius  of  its  base. 

4       3 

11.  Find  the  radius  of  gyration  of  a  concentric  spherical  shell  about 
a  tangent  to  the  external  sphere,  the  radii  being  a  and  b. 

12.  Find  the  radius  of  gyration  of  a  paraboloid  of  revolution  about 
its  axis,  in  terms  of  the  radius  {h)  of  the  base. 

3 

13.  Find  the  moment  of  inertia  of  an  eUipsoid  about  one  of  its 
principal  axes. 

15 


228  MECHANICAL   APPLICATIONS.  [Ex.  XV. 

14.  Find  the  radius  of  gyration  of  a  symmetrical  double  convex  lens 
about  its  axis,  a  being  the  radius  of  the  circular  intersection  of  tne 
two  surfaces,  and  b  the  semi-axis. 

15.  Find  the  radius  of  gyration  of  the  same  lens  about  a  diameter 
to  the  circle  in  which  the  spherical  surfaces  intersect. 

2o{b'  +  3«^)       ' 


THE  END. 


14  DAY  USE    Oyi 

SiAiiilicS  LIBRARY 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 

Renewed  books  are  subject  to  immediate  recall. 


i^-ffmi — 


^^  0  I   lUULk 


WOV  19,1965 


OCT 


Tm4^^ 


NOV  .2.a-1flg9 


1?f=^'M9B8 


-r-JiJNl     lO^efea." 


1963 


^ 


TTucrtrtJ 


^: 


,1 


1/5'" 


hii  21-50m-6,'60 
(jB1321slO)476 


General  Library 

University  of  California 

Berkeley 


YCI02 


* 


